From quarks to the universe a short physics course

The Nature of Physics

Modern-era Physics started as Natural Philosophy As this former name implies, Physics is built (and continues to be developed) around the age-old, yet ever-present questions:

• What the World and its parts are made of?How?

• Is there a hidden underlying simplicity in its immense complexity and diversity?

Physics explores both the natural and man-made world, from the smallest particles to the vast Universe, establishing itself as a foundational science that encompasses and supports other fields like Chemistry, Biology, and Engineering Its methodology is characterized by precision, quantitativeness, and mathematical abstraction, relying on observations and controlled experiments to generate ideas and validate or refute theories Furthermore, the formulation of fundamental quantitative relations is essential to the discipline, reinforcing Physics's critical role as the foundation for understanding the natural sciences.

E.N Economou, From Quarks to the Universe,

While science and engineering have made significant advancements, their influence is limited in complex areas like molecular and biological structures This gap allows for the continued growth and specialization of fields such as Chemistry, Biology, and Engineering.

Over the last 50 years or so Physics is actively concerned over another fundamental, age-old, but much more difﬁcult question which stretches its methodology to the limit:

• How did the World start, how did it evolve, and where is it going?

Observational data, such as the recession of distant galaxies and the distribution of Cosmic Microwave Background Radiation, combined with established physical theories, have enabled us to reconstruct key events in the Universe's history However, many significant occurrences, like the emergence of life, remain unknown and are under ongoing research Additionally, the development of a successful quantum theory of gravity is currently being explored, which aims to provide a concise description of the Universe's genesis.

The Subject Matter of Physics

The subject matter of Physics is summarized in Fig.1.1and Table1.1:

Ga laxi es Sup erc lust er of G ala xies

Sup erc lust er of G ala xies

Lar ger stru ctu res

This World, this Small World, the Great

Figure 1.1 illustrates the primary structures of matter, ranging from the smallest to the largest sizes in a clockwise direction, while also hinting at the potential connection between these two extremes The sizes are measured in meters, providing a clear scale for understanding the dimensions of various matter structures.

2 1 Introduction: The World According to Physics

Various Branches of Physics

In summary, this article highlights the various branches of Physics and their connections with specialized sciences, along with examples demonstrating the significant impact of Physics on key technologies.

Table 1.1 Levels of the structure of matter (see also [2])

Level of the structure of matter

Constituents Interaction(s) responsible for the structure

Quarks 10 −10 Atoms and/or ions and electrons E/ Μ

Molecules, cells, tissues, organs, microbes

Planets 10 6 –10 7 Solids, liquids, gases E/Μ, gravitational

Electrons, nuclei, ions, photons Gravitational, strong, weak, E/ Μ

White dwarfs 10 7 Nuclei, electrons Gravitational

Neutron stars 10 4 Neutrons and some protons and electrons

Galaxies 10 21 Stars, ordinary and dark matter, photons, neutrinos

Observable universe 10 26 Galaxies, dust, dark matter, dark energy

The Main Points of This Book: Basic Ideas Applied

to Equilibrium Structures of Matter 1

Protons and neutrons, which form the nucleus of an atom, are composed of elementary particles known as quarks; specifically, a proton is made up of two up quarks (u) and one down quark (d), while a neutron consists of two down quarks and one up quark Electrons (e) orbit these nuclei, creating the structure of atoms.

2 The constituents of all composite equilibrium structures are mutually attracted and self-trapped because of one or more of the four interactions presented in Table2.2.

The interactions within composite structures exert a continuous squeezing pressure, which is countered by the expanding pressure generated by the perpetual quantum motion of the constituent microscopic particles This quantum motion arises from the uncertainty principle and is influenced by the exclusion principle when multiple fermions are present Equilibrium in composite systems occurs when the squeezing pressure from interactions is precisely balanced by the expanding pressure from quantum motion This balance of pressures reflects the general principle of minimizing internal energy U under conditions of negligible external pressure and temperature.

Table 1.2 Connection of branches of Physics with Technologies and other Sciences and Mathematics

Nuclear physics Atomic and molecular physics Condensed matter physics

Atmospheric and space physics Meteorology, global climate Astrophysics, cosmology Astronomy

Solid state devices Integrated circuits Magnetic devices

X-rays c -rays Magnetic resonance (MRI) Positron annihilation (PET)

1 Section 1.4 summarizes the content of this book It may be useful for the reader to return to this section at later times.

Equilibrium of composite structures,Minimum of internal energyU Internal energy U= Internal potential energyEP + internal kinetic energyEK Internal quantum kinetic energy ofNidentical fermions:

E K ẳ2:87h 2 N 5 = 3 m V 2 = 3 ẳ1:105h 2 N 5 = 3 m R 2 non-relativistic Internal quantum kinetic energy ofNidentical fermions:

The interplay between the first and second laws of thermodynamics results in the Gibbs free energy equation, represented as G = U + PV - TS Under constant pressure and temperature, the Gibbs free energy consistently decreases as a system approaches equilibrium, ultimately reaching its minimum value at equilibrium When the contributions from PV and TS are minimal, G simplifies to U.

Dimensional analysis is an effective technique for deriving physics formulas by identifying the relevant parameters and universal physical constants that influence a quantity X The resulting formula for X is expressed as a product of appropriate powers of typically three of these parameters or constants, combined with a function of their dimensionless combinations, ensuring that the overall dimensions align with those of X.

Next, we apply the general principles presented above to several basic equilibrium structures; the main properties of them are expressed in terms of universal physical constants.

(i)Nucleiconsist ofZprotons andNneutrons, i.e ofA=Z+Nnucleons Both the strong interactions and the Coulomb interactions contribute to the potential energy: EPẳ 1 2ANnnVsỵ 1 2

The equation P ije 2 =rij describes the interaction between nucleons, where Nnn represents the number of nearest neighbors and −Vs signifies the strong force acting between them Additionally, the double summation in the Coulomb term accounts for interactions among all Z protons The radius of the nucleus, R, is proportional to the cube root of the mass number A, while the total kinetic energy encompasses both the kinetic energy of protons and neutrons.

Atoms are composed of a nucleus containing Z protons, with Z electrons surrounding it due to Coulomb interactions The ground state energy of these electrons is estimated by calculating single electron atomic orbitals and their energy levels According to Pauli’s principle, electrons fill the lower energy levels first, continuing until all Z electrons are accounted for.

Molecules are formed from various combinations of atoms, cations, and anions, which are held together by Coulomb interactions The structure of molecular orbitals can be described as linear combinations of atomic orbitals or through hybridized atomic orbitals.

In solids and liquids, numerous atoms and molecules interact through Coulomb forces In metals, the primary kinetic energy is attributed to detached electrons, estimated by the formula EK ≈ 2.87 m h² N^(5/3) eV^(2/3) Meanwhile, the potential energy, characterized by Coulomb interactions, is expressed as EP ≈ N a Eo c/r.

Na is the total number of atoms,Eoẳe 2 =aB,aBh 2 =mee 2 is the so-called Bohr radius, r is connected to the volume by the relation 4p 3 ðr a B ị 3 ẳ ðV=N a ị, c

0:56f 4 = 3 þ0:9f 2 ; and fis the valence For semiconductors and insulators, as for molecules, a linear combination of atomic (hybridized or not) orbitals turns out to be a more convenient way of studying them.

(v) Planets are spherical objects of mass MẳN m u and radius R, whereN m ẳ

The total number of nucleons in all nuclei is represented by AWNa 10 49 –10 55, with AW indicating the average atomic weight In planetary bodies, both Coulomb and gravitational interactions contribute to potential energy, while kinetic energy primarily arises from electrons, similar to solids and liquids The spherical shape of moons, planets, and stars results from the long-range nature of gravitational interaction, which is significant due to the immense mass involved.

Dead stars are classified into three types: white dwarfs, neutron stars, and black holes, with masses relative to the Sun's mass (MS) White dwarfs have a radius similar to Earth's, being approximately 100 times smaller than their previous active star phase This significant compression leads to the detachment of electrons from their parent atoms, resulting in a composition of electrons and bare nuclei The total energy is minimized, with the kinetic energy primarily from electrons and gravitational potential energy, leading to a relationship between radius and mass expressed as R = 1.42 h².

As the mass of a white dwarf increases, the kinetic energy of its electrons approaches an extreme relativistic limit, expressed as A/R, akin to the gravitational potential energy represented by -B/R When the gravitational force surpasses the electron degeneracy pressure (B > A), the white dwarf will ultimately collapse into a neutron star This transition occurs at a critical mass threshold, defined by the equality B = A, which corresponds to a critical value of N m; cr1 = 0.77 Gu h c^2.

In a neutron star, the collapse of matter causes electrons to be confined within atomic nuclei, resulting in a composition primarily of neutrons, with a small fraction of protons and electrons The gravitational potential energy dominates, while the kinetic energy is largely contributed by neutrons and, to a lesser extent, protons, with electrons exhibiting extreme relativistic speeds By minimizing the total energy concerning the radius, we find that the optimal radius is approximately 3.16 h².

Neutron stars have a radius significantly smaller than that of white dwarfs, typically by a factor of about \(10^{-3}\) As the mass of a neutron star increases, the kinetic energy of its neutrons and protons reaches extreme relativistic levels When the condition \(A = B\) is met, the neutron star will ultimately collapse into a black hole, with the critical mass defined by the equation \(N m_{cr} \approx 21.6 \frac{G m h}{c^2 n}\).

3 = 2 which corresponds to about 3MS. Both the minimum and the maximum mass ofactive starscan be expressed in terms of physical constants:N m; min 0:2 m u e

The Universe is expanding, as evidenced by observational data and the general theory of relativity This expansion is characterized by an increasing distance (R) between two distant points, which is proportional to R itself, expressed mathematically as Ṙ = H × R, where H represents the Hubble constant The fundamental relationship governing this expansion can be summarized by the equation Ṙ/R = H.

The total average energy density of the Universe, denoted as e, is composed of various contributions, including photons, neutrinos, baryons, dark matter, and dark energy The energy density from photons is inversely proportional to the fourth power of the scale factor (1/R^4), while the contributions from neutrinos, baryons, dark matter, and dark energy also play significant roles in the overall energy balance of the cosmos.

1=R 3 ), the fourth to dark matter (proportional to 1=R 3 ), and theﬁfth one to the dark energy (which seems to be a constant independent ofR).

1 S.W Hawking, L Mlodinow, The Grand Design (Bantam Books, NY, 2010)

2 P Morrison, P Morrison, Powers of Ten (Scienti ﬁ c American Books, NY, 1982)

Three Key-Ideas and a Short-Cut

Introduction

In conclusion, this article highlights the diverse branches of Physics and their connections with specialized sciences, while also showcasing the significant influence of Physics on key technologies.

Table 1.1 Levels of the structure of matter (see also [2])

Level of the structure of matter

Constituents Interaction(s) responsible for the structure

Quarks 10 −10 Atoms and/or ions and electrons E/ Μ

Molecules, cells, tissues, organs, microbes

Planets 10 6 –10 7 Solids, liquids, gases E/Μ, gravitational

Electrons, nuclei, ions, photons Gravitational, strong, weak, E/ Μ

White dwarfs 10 7 Nuclei, electrons Gravitational

Neutron stars 10 4 Neutrons and some protons and electrons

Galaxies 10 21 Stars, ordinary and dark matter, photons, neutrinos

Observable universe 10 26 Galaxies, dust, dark matter, dark energy

1.4 The Main Points of This Book: Basic Ideas Applied to Equilibrium Structures of Matter 1

1 Out of the elementary matter-particles presented in Chapter 2, Table2.1only theup quark(u) and thedown quark(d) make up the proton consisting of two u’s and one d and the neutron consisting of two d’s and one u.Electrons(e ) are trapped around nuclei, made of protons and neutrons, to form atoms.

2 The constituents of all composite equilibrium structures are mutually attracted and self-trapped because of one or more of the four interactions presented in Table2.2.

3 The interactions, which tend to continuously squeeze the composite structures, are counterbalanced by the pressure due to the perpetual motion of the constituents microscopic particles This motion is of quantum nature and stems from the uncertainty principle aided by the exclusion principle (if more than two fermions (see Sect.3.3) are involved) In other words, equilibrium of composite systems is established when the squeezing pressure of the interactions is exactly balanced by the expanding pressure of the quantum perpetual motion of the constituent particles The equality of pressures is a consequence of the general principle of the minimization of the internal energy U (under conditions of negligible external pressure and temperature) Thus:

Table 1.2 Connection of branches of Physics with Technologies and other Sciences and Mathematics

Nuclear physics Atomic and molecular physics Condensed matter physics

Atmospheric and space physics Meteorology, global climate Astrophysics, cosmology Astronomy

Solid state devices Integrated circuits Magnetic devices

X-rays c -rays Magnetic resonance (MRI) Positron annihilation (PET)

1 Section 1.4 summarizes the content of this book It may be useful for the reader to return to this section at later times.

Equilibrium of composite structures,Minimum of internal energyU Internal energy U= Internal potential energyEP + internal kinetic energyEK Internal quantum kinetic energy ofNidentical fermions:

E K ẳ2:87h 2 N 5 = 3 m V 2 = 3 ẳ1:105h 2 N 5 = 3 m R 2 non-relativistic Internal quantum kinetic energy ofNidentical fermions:

4 The combination of theﬁrst and the second law of thermodynamics leads to the following relation: The so-called Gibbs free energy,GUþPVTS;under conditions of constant pressure and temperature, is always decreasing during the system’s path towards equilibrium and reaches its minimum value when equilibrium is established.Greduces toUwhenPVandTSare negligible.

5 Dimensional analysis is a powerful method for producing physics formulae It requires the identiﬁcation of the parameters and/or the universal physical constants on which a quantityXmay depend Then the formula forXis a product (with the same dimensions asX) of appropriate powers of usually three of those parameters/physical constants times a function of their dimensionless combinations.

Next, we apply the general principles presented above to several basic equilibrium structures; the main properties of them are expressed in terms of universal physical constants.

(i)Nucleiconsist ofZprotons andNneutrons, i.e ofA=Z+Nnucleons Both the strong interactions and the Coulomb interactions contribute to the potential energy: EPẳ 1 2ANnnVsỵ 1 2

In nuclear physics, the energy of a nucleus can be described using the equation P ije 2 = rij, where Nnn represents the number of nearest neighbor nucleons The strong interaction, denoted as -Vs, occurs between pairs of these nearest neighbor nucleons, while the double summation in the Coulomb term accounts for all Z protons within the nucleus Additionally, the radius of the nucleus (R) is proportional to the one-third power of the mass number (A), and the total kinetic energy consists of the kinetic energies of both protons and neutrons.

Atoms are composed of a nucleus containing Z protons, with Z electrons held in orbit around it through Coulomb interactions The ground state energy of these electrons is estimated by calculating the single electron atomic orbitals and their associated energy levels Electrons fill these energy levels starting from the lowest, adhering to Pauli’s principle until all Z electrons are allocated.

Molecules are formed from atoms, cations, and anions, which can combine in virtually limitless ways through Coulomb interactions The structure of molecular orbitals can be effectively represented as linear combinations of atomic orbitals or through hybridized atomic orbitals.

In solids and liquids, a vast number of atoms and molecules interact through Coulomb forces In metals, the primary source of kinetic energy is attributed to detached electrons, which can be estimated using the formula EK ≈ 2.87 m h² N^(5/3) eV^(2/3) Meanwhile, the potential energy is characterized by a Coulombic form, expressed as EP ≈ N a Eo c/r.

Na is the total number of atoms,Eoẳe 2 =aB,aBh 2 =mee 2 is the so-called Bohr radius, r is connected to the volume by the relation 4p 3 ðr a B ị 3 ẳ ðV=N a ị, c

0:56f 4 = 3 þ0:9f 2 ; and fis the valence For semiconductors and insulators, as for molecules, a linear combination of atomic (hybridized or not) orbitals turns out to be a more convenient way of studying them.

(v) Planets are spherical objects of mass MẳN m u and radius R, whereN m ẳ

AWNa 10 49 –10 55 is the total number of nucleons within all nuclei, AW is the average atomic weight, andu12 1 mðC 12 ị In planets both the Coulomb interaction as well as the gravitational one, EGẳ aGGM 2 =R; contribute to the potential energy, while the kinetic energy is due mainly to the electrons as in solids and liquids The spherical shape of moons, planets and stars is a consequence of the long range character of the gravitational interaction which becomes appreciable as a result of the huge mass involved.

Dead stars are classified into three types: white dwarfs, neutron stars, and black holes, with their masses ranging from less than 1.4 times to over 3 times the mass of the Sun White dwarfs are approximately 100 times smaller in radius than their previous active star phase, comparable to Earth's size This significant compression results in a composition of electrons and bare nuclei, as all electrons are detached from their parent atoms The primary source of kinetic energy in white dwarfs is the electrons, while gravitational potential energy plays a crucial role The relationship between the radius and mass of a white dwarf can be expressed mathematically, indicating that the radius is proportional to the mass raised to the power of 1.42.

Gu 2 m e N m 1=3 ,MẳN m u If the mass of the white dwarf keeps increasing, the kinetic energy of the electrons tends to the extreme relativistic limit which is of the formA/R, i.e similar to the gravitational one,−B/R Thus whenBAthe white dwarf will collapse to a neutron star Hence, the equalityBẳAgives the collapse critical value, which is N m; cr1 ẳ0:77 Gu h c 2

In a neutron star, the collapse leads to electrons being confined within the nuclei, resulting in a composition primarily of neutrons (approximately 93.4% of the mass) and a small fraction of protons and electrons (around 6.6%) The gravitational potential energy dominates, while the kinetic energy is mainly contributed by neutrons and, to a lesser extent, protons, with electrons exhibiting extreme relativistic behavior By minimizing the total energy concerning the radius, we find that the optimal radius is approximately 3.16 times h squared.

The radius of a neutron star is significantly smaller than that of a white dwarf, approximately by a factor of \(10^{-3}\) As the mass of a neutron star increases, the kinetic energy of neutrons and protons reaches extreme relativistic levels, leading to a condition where the star can collapse into a black hole This collapse occurs when the balance condition, represented as \(A = B\), is met, which can be expressed mathematically as \(N m_{cr} \approx 21.6 \frac{G m h}{c^2 n}\).

3 = 2 which corresponds to about 3MS. Both the minimum and the maximum mass ofactive starscan be expressed in terms of physical constants:N m; min 0:2 m u e

The Universe is expanding, as evidenced by observational data and the general theory of relativity This expansion is characterized by an increasing distance (R) between distant points, which grows at a rate proportional to R itself, expressed by the equation Ṙ = H * R, where H is known as the Hubble constant.

The Elementary Particles of Matter

Each m-particle listed in Table 2.1 has a corresponding m-antiparticle, indicated by a bar above the particle's symbol For instance, the antiparticle of the e-neutrino is denoted as some, while the down quark's antiparticle is represented as d Historically, the electron's antiparticle is called the positron, denoted as e⁺ Antiparticles possess the same rest mass and spin as their corresponding particles but have opposite electric charge, lepton number, and baryon number Notably, antiparticles do not contribute to the structure of matter, as a particle-antiparticle pair annihilates upon contact, producing photons or, less commonly, neutrinos Despite their brief existence, antiparticles can be artificially generated, and their properties have been experimentally verified Quarks have not been observed as isolated free particles; instead, they exist only within composite structures like baryons (protons and neutrons) or mesons, leading to uncertainties in accurately determining their rest mass, particularly for the lighter quarks.

Quarks possess a unique type of charge known as color-charge (c-charge), in addition to their electric charge This c-charge is responsible for the strong nuclear force, similar to how electric charge governs the electromagnetic force Unlike electric charge, which has only one type, c-charge comes in three varieties: red (R), green (G), and blue (B) The corresponding opposite charges are referred to as antired, antigreen, and antiblue, denoted as R̅, G̅, and B̅ Each quark carries one of the three c-charges, while each antiquark carries the corresponding c-anticharge The interaction between quarks and antiquarks forms the basis of various particle combinations in quantum chromodynamics.

Positron Emission Tomography (PET) is a medical diagnostic technique that involves introducing positron-emitting substances into a patient's body When positrons encounter electrons, they annihilate, producing two photons that travel in opposite directions, each carrying an energy of 0.511 MeV, equivalent to the rest mass of electrons.

5 Mesons are composite short-lived entities consisting of one quark and one antiquark Baryons and mesons are collectively called hadrons.

14 2 The Atomic Idea of the formRR, orGG, orBBhas zero c-charge (in other words, can be considered as colorless) Similarly colorless is the combination of three quarks of c-chargesR,

G, andBrespectively or of three antiquarks of c-antichargesR; G;Brespectively.

The notation R, G, and B for the three types of c-charge is appropriate, as it parallels Newton’s disc, where the equal mix of red, green, and blue light results in white By extending this analogy, we can describe an antiblue c-charge as being equivalent to the absence of blue in this color combination.

RGcombination of c-charges, could be presented as a“yellow”c-charge [2].

Out of the twelve elementary m-particles and their corresponding m-antiparticles, only three— the electron, up quark, and down quark— are essential for the structure of all ordinary matter that exists in our surroundings and within us.

The Interactions and Their Elementary Interaction-Carrying-Particles

The four fundamental interactions in physics are gravitational, electromagnetic, weak nuclear, and strong nuclear forces These forces operate through the exchange of intermediate particles (ic-particles), where an m-particle emits an ic-particle that is then absorbed by another m-particle, resulting in a force between them This process can also lead to the transformation of m-particles and the creation of m-particle/m-antiparticle pairs Each of these interactions adheres to strict conservation laws, ensuring that specific quantities, such as electric charge, momentum, angular momentum, color charge, baryon number, and lepton number, remain constant before and after the interactions Additionally, total energy must be conserved throughout these processes, aligning the energy of initial and final particles Certain interactions, like the strong force, also uphold extra conservation laws, which significantly restrict the types of physical processes that can occur.

In Table 2.2 we present some properties of the four interactions and of their elementary ic-particles.

6 Neutrinos of a given family (see Table 2.1) as they travel after leaving a reaction exhibit a transformation from one family to another; e.g an e-neutrino may change to a μ -neutrino and vice versa.

Table 2.2 outlines the four fundamental interactions in physics, detailing their interaction-carrying particles Gravitational interactions, characterized by a dimensionless strength of G and a range of approximately 10^39 meters, involve hypothetical gravitons with zero rest mass and spin, affecting all matter and interaction-carrying particles Electromagnetic interactions, with a strength of 1/137, feature photons as their carriers, which can be either attractive or repulsive and also possess zero rest mass and spin, influencing electrically charged particles The weak nuclear interaction is represented by vector bosons, with a strength of around 10^5 and a range of 10^-18 meters, interacting primarily with up and down quarks.

All particles, including photons and vector bosons, exhibit interactions that can be either attractive or repulsive Strong nuclear forces, mediated by gluons, play a crucial role in the behavior of quarks and gluons These electromagnetic interactions lead to the formation of neutral systems, where the residual electromagnetic forces become short-range, typically on the order of a few angstroms.

The symbols in the second column of the table include G, representing the gravitational constant; mp, denoting the rest mass of the proton; h, which stands for the reduced Planck's constant, a hallmark of Quantum Mechanics; c, the speed of light in a vacuum; gw, indicating the strength of the weak interaction; and mw, the mass of the weak boson.

The W boson and the strong interaction strength, denoted as gs, play crucial roles in particle physics The range of each interaction is related to the rest mass (m) of the corresponding particle through the formula r = h/mc Notably, the strong interaction presents unique characteristics, which will be explored in Chapter 7 For detailed numerical values of these quantities, refer to Table I.1 in Appendix I.

Table 2.2 highlights that the strength of gravitational interaction is vastly weaker than other fundamental forces For instance, when lifting an object like a chair, the electric force from your muscles surpasses the gravitational pull of the entire Earth on that chair Despite its relative weakness, gravity becomes significant in large-scale systems, such as planets and stars, where it ultimately dominates due to the immense number of particles involved.

The N m-particles in the system interact gravitationally with one another due to gravity's long-range effects Each pair of particles contributes positively to the gravitational self-energy of the system, as all interactions are attractive in nature.

Gravitational waves and their associated particles, known as gravitons, remain undetected in experiments due to their extremely low energy levels It is important to note that all particles, whether massive or massless, interact with gravitational forces as they possess energy and relativistic mass.

The electromagnetic (E/M) interaction is crucial in shaping matter, influencing scales from atoms to asteroids, spanning 15 orders of magnitude Its unique particle, the photon, is exceptional as it not only mediates force but also travels freely through space, transmitting energy and information over vast distances Unlike other particles, such as gravitons, which are undetectable with current technology, vector bosons that decay rapidly, and gluons that remain confined within composite structures, photons can be easily emitted and detected These distinctive characteristics highlight the significant biological and technological roles of photons, warranting further exploration in Chapter 6.

Feynman Diagrams

Interactions in physics not only mediate forces but also transform particles and create pairs of particles and antiparticles Feynman diagrams effectively illustrate these processes, offering a clear visual representation of ongoing physical interactions While they also enable complex quantum mechanical calculations through intricate rules, this article will focus solely on the descriptive aspects of Feynman diagrams.

In Fig 2.2 we employed a Feynman diagram to present the electromagnetic interaction between two particles carrying electric charge; this interaction is mediated by the emission and subsequent absorption of a photon 7

In Fig.2.3we present through Feynman diagrams processes involving emission or absorption of photons by an electron or processes where an electron/positron pair appears or disappears.

The weak interaction plays a crucial role in the transformation of quarks and leptons, facilitated by the emission of charged vector bosons (W⁺, W⁻) and the electrically neutral Z⁰ boson These emitted particles quickly decay into pairs of quark/antiquark or lepton/antilepton It's important to note that all massive particles, including their antiparticles, as well as photons and vector bosons, are influenced by weak interactions The significant mass of vector bosons, as highlighted in Table 2.2, accounts for the weak force's extremely short range.

In a basic Feynman diagram, the interaction between an electron and a proton, two electrically charged particles, is illustrated through the exchange of a photon This process involves the electron emitting a photon, which is then absorbed by the proton, or the reverse scenario This interaction is linked to the potential energy, denoted as V, which can be roughly described by the established Coulomb formula.

The equation V = e²/r describes the interaction between the electric charges of protons and electrons, where e represents the charge of the proton and the charge of the electron, and r is their distance, measured at approximately 0.53 x 10^-10 m In this scenario, the energy value is 27.2 eV, indicating that the photon does not manifest in either the initial or final state, categorizing it as a virtual particle, specifically a virtual photon.

In this book, the Gauss-CGS (G-CGS) system is preferred for electromagnetic (E/M) quantities due to its clarity, as it explicitly incorporates the velocity of light in Maxwell's equations Conversely, the more commonly used SI system offers familiar and convenient units For a comparison of the quantities in SI and G-CGS, refer to Appendix B, which includes key relationships such as e0 = 1/4π, l0 = 4π/c², and the formulas for electric field (E), displacement field (D), magnetic field (B), and magnetic field intensity (H) Additionally, the Coulomb potential in SI is expressed as e² = 4πε₀r, and widely used formulas will also be provided in SI units.

The atomic interaction described by the formula \(0 \rightarrow h = m c\) indicates the range of interaction In the process illustrated in Fig 2.4, a quark is transformed into another quark through the emission or absorption of a W vector boson, which subsequently decays into an electron and an electron-antineutrino This process, known as beta decay, converts a neutron (u, d, d) into a proton (u, u, d) while emitting an electron and an electron-antineutrino Beta decay occurs in the byproducts of nuclear reactors, contributing to their radioactivity Often, this decay is followed by the emission of a highly energetic photon, referred to as gamma radioactivity, as the radioactive nucleus typically remains in an excited state post-beta decay.

In Fig.2.5the Feynman diagram for the process of a muon decay to an electron, an e-antineutrino, and a l-neutrino taking place through the weak interaction is

The processes involving a free photon include several key interactions: a photon is emitted by a decelerating electron, which is essential in X-ray production; an electron absorbs a photon, as observed in the photoelectric effect; a high-energy photon can annihilate to create an electron-positron pair; an electron-positron pair can annihilate, resulting in the emission of two photons, a principle used in PET diagnostics; and an electron can absorb one photon and emit another, as seen in Raman scattering Notably, antiparticles like positrons are represented in these diagrams with arrows indicating a direction opposite to the flow of time.

Beta decay is a fundamental process where a down quark is converted into an up quark, resulting in the transformation of a neutron into a proton, along with the emission of an electron and an electron antineutrino This process contributes to the radioactivity of nuclear reactor byproducts, as one of the neutrons in the daughter nucleus undergoes this transformation The weak interaction, represented by wavy lines in Feynman diagrams, facilitates this decay through the emission and absorption of a W boson In these diagrams, antiparticles like the electron antineutrino are indicated by arrows pointing in the opposite direction of time Additionally, the transformation of an up quark and an electron into a down quark and an electron neutrino, via the exchange of a charged vector boson, occurs in certain heavy nuclei, where a proton is converted into a neutron through the absorption of an inner orbiting electron and the emission of an electron neutrino.

In Feynman diagrams, interactions at each vertex require the conservation of specific quantities, including momentum, angular momentum, electric charge, color charge, baryon number, and lepton number for each family For every vertex, the total baryon number must remain constant, meaning the baryon number before the interaction equals the baryon number after While energy conservation is not necessary at each elementary vertex, it must be conserved overall, ensuring that the total energy before a reaction matches the total energy after it concludes.

Strong interactions are mediated by particles known as gluons, of which there are eight distinct types based on their combinations of c-charge and c-anticharge Out of these, six gluons possess a net c-charge, represented by the combinations RG, RB, GR, GB, BR, and BG.

A muon decays into a muon neutrino by emitting a W vector boson, which then decays into an electron and an electron antineutrino This decay process can also occur when a W+ vector boson is created alongside an electron and an electron antineutrino, leading to the annihilation of the muon and the production of a muon neutrino According to the principle of energy conservation, the rest energy of the muon must equal the total of the rest energies of the electron, muon, and muon neutrino, along with their kinetic energies in the system where the muon is at rest.

A u quark can be transformed into a d quark through the emission of a W⁺ vector boson, which subsequently interacts with an existing electron, resulting in the disappearance of the W⁺ and electron pair and the creation of an electron neutrino (νₑ) This process can also occur when the electron emits a W boson, which is then absorbed by the u quark, facilitating its transformation into a d quark.

20 2 The Atomic Idea them carry the same c-charge/c-anticharge and as such are colorless: ðRRGGị= ffiffiffi p2

Gluons are responsible for binding three quarks together to form baryons, such as protons and neutrons, which are colorless due to their different color charges A proton consists of two up quarks and one down quark, while a neutron is made up of two down quarks and one up quark Free protons are stable, with no observed decay, although some theoretical models suggest they may have an extremely long lifetime In contrast, free neutrons have a mean lifetime of 889 seconds but remain stable within non-radioactive nuclei Additionally, there are several other short-lived baryons, all of which are colorless.

Quarks and antiquarks can form short-lived particles known as mesons through gluon exchange, and these mesons are colorless The simplest mesons, made up of a quark and an antiquark from the first family, are referred to as pions There are three varieties of pions: positively charged (p⁺), negatively charged (p⁻), and neutral (p⁰).

Concluding Comments

The Standard Model of particle physics posits the existence of the Higgs particle, which is essential for explaining the mass of elementary particles through virtual interactions This particle is characterized by having zero spin and zero electric charge, with an expected rest energy between 114,000 and 158,000 MeV Due to its significant rest mass, producing the Higgs particle requires extremely high energy levels In 2012, the Large Hadron Collider (LHC) at CERN successfully detected the Higgs particle, confirming its rest energy at 125,000 MeV, thereby providing strong evidence supporting the Standard Model.

Despite the Standard Model's success in explaining elementary and composite particle interactions, it leaves several profound questions unanswered These include the reasons for the existence of only three families of elementary particles, the factors determining their rest masses, and the origins of the observed values of interaction strengths Additionally, there is speculation about a deeper level of understanding at a smaller scale that could address these inquiries Numerous theoretical frameworks have been proposed to extend beyond the Standard Model, aiming for a more unified perspective, often introducing complex concepts and unfamiliar mathematics One notable approach in this pursuit is supersymmetry.

The theory predicts the existence of additional elementary particles, suggesting one extra fermion for each boson in the Standard Model and vice versa It also indicates a unification energy where all interaction coupling constants converge In contrast, string theory proposes that the fundamental components of the universe are not point-like particles but rather extended objects like strings and membranes, existing at a scale of 10^-35 meters—far beyond current experimental capabilities Despite the absence of experimental evidence, string theory has been a focus of research for the past 30 years due to its intriguing characteristics.

The hierarchical structure of matter begins with the combination of three quarks (u, u, d) or (u, d, d) via gluon exchange, resulting in the formation of protons and neutrons These particles then attract each other through strong interactions to create atomic nuclei The positively charged protons and other composite nuclei draw in negatively charged electrons, leading to the formation of electrically neutral atoms Atoms further interact through electrical forces to create a vast array of molecules, ultimately giving rise to living systems and various astronomical objects, all held together by gravitational forces A key question arises: why do these attractive interactions not lead to infinite compression, as seen in the gravitational collapse of massive stars into black holes? What mechanisms transform attraction into repulsion when particles are pressed together, maintaining stability in physical systems?

We thus returned to where we started, namely to the third question posed in the summary of this chapter:

What counterbalances the overall attraction of the interactions and establishes equilibrium?

Feynman suggested that atoms are "little particles that move around in perpetual motion," which creates a pressure that can potentially destabilize the system When this pressure equals the attractive forces' squeezing pressure, the system reaches equilibrium This raises important questions about the origin of perpetual motion and the physical mechanisms behind it The answers lie in the fundamental yet challenging concept of particle-wave duality, which will be explored in the next chapter.

Summary of Important Concepts, Relations, and Data

This chapter introduces fundamental concepts in particle physics, including elementary m-particles and ic-particles, as detailed in Tables 2.1 and 2.2 Key topics covered include protons, neutrons, various baryons, mesons, antiparticles, electric charge, color charges, and Feynman diagrams Additionally, it discusses the relationship between the range of an interaction (r) and the mass of the corresponding ic-particle (m), expressed by the equation r = h / (m c).

The exceptional case of strong interactions will be discussed in a later chapter.

Multiple-Choice Questions/Statements

The correct answers to the multiple choice questions are given at the end of the book (see Appendix H).

Solved Problems

1 Draw the Feynman diagrams for the decays ofp 0 andp þ and estimate the mean lifetime of these pions What is the energy and the momentum of the decay products?

The zero pion, composed of a quark and its corresponding antiquark, annihilates to produce two photons, as illustrated in the Feynman diagram When at rest, a zero pion cannot decay into a single photon without violating momentum conservation, necessitating the emission of two photons with equal energies of approximately 67 MeV each The mean lifetime of such decay processes is influenced by the complexity of the Feynman diagram and the interaction type; generally, stronger interactions lead to shorter lifetimes In the case of the electromagnetic interaction depicted, the expected mean lifetime is around 10^-16 seconds, with the actual measurement being approximately 8.4 x 10^-17 seconds.

The decay of the proton (p⁺), composed of an up quark and an anti-down quark, does not involve photons or gluons due to charge conservation Instead, the decay process is mediated by the emission of a W⁺ vector boson, which subsequently creates a lepton pair with zero total lepton number and a charge of +1, such as a positron and an electron neutrino or an antimuon and a muon neutrino Notably, the decay process p⁺ → l⁺ + νₗ is significantly more probable Additionally, a similar process can occur with the creation of an antimuon and muon neutrino pair, facilitated by the absorption of the W boson by the incoming proton The mean lifetime of the proton is expected to be at least six orders of magnitude longer than that of the neutral pion, as both lifetimes involve two vertices, each contributing to the overall lifetime.

24 2 The Atomic Idea inversely proportional to the ratio of the dimensionless strengths of the interactions responsible for their decay The mean lifetime ofp þ is actually 2:610 8 s.

2.The range r of the weak interaction is about2:210 18 m What is the rest energy of the vector bosons?

Solution From the formularẳh=m cẳh c=m c 2 it follows thatm c 2 ẳh c=r We shall work in the atomic system of units wherehẳeẳm e ẳ1 (see Table I.2 in Appendix I).

In this system, the speed of light in a vacuum is approximately 137, which is the inverse of the fine structure constant The Bohr radius, measuring 0.529 Å (where 1 Å = 10^-10 m), serves as the unit of length, while the unit of energy is defined as h² = mea²B = 27.2 eV The range r in atomic units is calculated to be about 2.2 × 10^18 / 0.529 × 10^-10 ≈ 4.16 × 10^8 Consequently, the rest energy of the vector boson in atomic units is given by m c² = h c/r = (1 - 137/4.16) × 10^8, resulting in approximately 32.93 × 10^8 atomic units of energy, equivalent to 89,600 MeV Actual measured values for the rest energy of vector bosons are 91,000 and 80,000 MeV, as indicated in Table 2.2.

Unsolved Problems

1 Create a table giving the symbol, the rest energy, the mass, the spin, and the electric charge of all leptons and all quarks.

2 Create a table giving the names, the dimensionless strengths, the range, the corresponding ic-particles (their rest energies, their spin, their electric charge and their emitters and absorbers) of all four basic interactions.

1 S.W Hawking, L Mlodinow, The Grand Design (Bantam books, New York, 2010)

2 M Veltman, Facts and Mysteries in Elementary Particle Physics (World Scienti ﬁ c, NewJersey, 2003)

The wave-particle duality concept posits that all entities, despite their particle-like nature, exhibit wave-like behavior, forming the foundation of Quantum Mechanics This duality provides the necessary kinetic energy to counteract attractive interactions, stabilizing both microscopic and macroscopic structures of matter The significance of Planck's constant in equations related to these structures underscores this principle Quantum Mechanics is fundamentally anchored in three core principles articulated by Heisenberg, Pauli, and Schrödinger, which are essential for understanding the subject matter that follows.

Concepts and Formulae

A classical particle is defined by two key properties: its composition, which can be either elementary and indivisible or made up of bound indivisible entities, and its motion, which follows a well-defined trajectory.

Classical waves exhibit two key properties: they are continuous and infinitely divisible, allowing them to overlap and interfere with other waves According to the principle of wave-particle duality, all entities, whether viewed as particles or waves, possess discrete characteristics while not adhering to a defined trajectory.

Throwing two stones into a calm pond creates a clear illustration of interference, as the resulting circular waves interact by canceling each other at certain points while reinforcing at others.

Particles exhibit a dual nature, behaving both as particles and waves, which challenges our everyday perceptions Despite initial skepticism, this concept has gained widespread acceptance due to extensive experimental evidence and its remarkable quantitative explanatory capabilities Recently, renewed interest in the foundations of Quantum Mechanics has emerged, driven by new theoretical insights and technological advancements, including quantum cryptography and the potential for quantum computing.

Wave-particle duality suggests that a particle of energy \( e \) and momentum \( p \) does not follow a classical trajectory but instead propagates as a wave characterized by angular frequency \( x = \frac{e}{h} \) and wavevector \( k = \frac{p}{h} \), where \( h \) is the reduced Planck’s constant This concept redefines the particle as a wave-particle, indicating that it is not a classical particle but rather a wave composed of discrete, indivisible entities of energy and momentum, given by \( e = hx \) and \( p = hk \).

Quantum Mechanics presents a unified theory that merges the seemingly opposing ideas of particles and waves, giving rise to the term "wave-particle." This theory is built upon three core principles: uncertainty, the exclusion principle for fermions, and spectral discreteness when the wave-particle is confined.

The first two principles quantitatively explain the perpetual microscopic motion needed to counteract attractive forces and maintain equilibrium structures in matter The third principle allows composite microscopic structures to function similarly to elementary ones, up to a certain limit, demonstrating a remarkable balance between stability and the necessity for change.

Wave-particle duality, despite initial skepticism regarding its paradoxical nature, has been firmly established as a fundamental principle of Nature, supported by extensive empirical evidence This concept reveals that particles exhibit both wave-like and particle-like properties, reshaping our understanding of the physical world.

The wavevector's magnitude \( k \) is connected to the wavelength \( \lambda \) through the relation \( k = \frac{2\pi}{\lambda} \), with its direction indicating the wave's propagation Entities in the universe possess both particle-like qualities, such as discreteness and indivisibility, and wave-like characteristics, including non-locality essential for interference phenomena Hence, these entities are referred to as wave-particles or particle-waves Feynman's gedanken two-slit experiment illustrates the wave-particle duality, demonstrating that classical laws of motion do not apply to microscopic particles Experimental evidence reveals that wave-particles display interference, supporting the concept of wave-particle duality The implications of this duality and Quantum Mechanics extend beyond the microscopic realm, affecting the properties of all structures in the universe, as the atomic theory links the characteristics of any matter to the behavior of its microscopic constituents.

We would like to stress this point emphatically.

The Properties of the Structures of the World at Every

The significance of the universal constant \( h \) in Quantum Mechanics (QM) is evident when examining macroscopic properties like density, compressibility, and specific heat, as well as the fate of stars—whether they become white dwarfs, neutron stars, or black holes If \( h \) were zero, the current structures of the universe would collapse into classical black holes While Classical Mechanics (CM) can yield accurate results in certain scenarios, such as the specific heat of insulators at high temperatures and the motions of sea waves or planetary orbits, this does not indicate a failure of QM; rather, QM simplifies to CM in these instances Furthermore, establishing key aspects of QM, such as coherence for wave interference and the unique correlations in entangled systems, becomes increasingly complex when dealing with multiple microscopic particles.

3.1 Concepts and Formulae 29 associated with entangled systems wereﬁrst discussed by Einstein-Podolsky-Rosen in relation with a gedanken experiment [2] which was later realized by Aspect et al.

Modern textbooks on Quantum Mechanics explore the complexities of the macroscopic world, emphasizing that it is not purely classical A truly classical world, characterized by a lack of quantum effects, would lack the intricate structures we observe today.

We have already mentioned that QM synthesizes the contradictory concepts of wave and particle but at a cost which is twofold:

The article discusses a probabilistic approach to understanding the world, utilizing the concept of a wavefunction, denoted as w(r,t), which indicates the probability density of locating a wave-particle at a specific position r and time t Unlike classical physics, which follows a single trajectory, this framework considers infinitely many possible trajectories connecting an initial point A to a final point B The probability of transitioning from A to B is determined by calculating the square of the absolute value of the sum of the probability amplitudes associated with these numerous trajectories.

Quantum Mechanics (QM) requires advanced mathematical concepts that go beyond typical university calculus However, its remarkable explanatory power, which spans from the structure of protons and neutrons to the entire universe, justifies this complexity Fortunately, much of this explanatory strength can be retained with a simpler approach This book will demonstrate how to focus on three fundamental principles of QM—Heisenberg, Pauli, and Schrödinger—derived from wave-particle duality, while also utilizing dimensional analysis to simplify understanding.

In the foundational framework of non-relativistic Quantum Mechanics, the time-dependent Schrödinger equation plays a crucial role in defining the wave function w(r, t) based on the potential energy V(r, t) This equation describes how the future state of the system, represented by the wave function, evolves over time, illustrating the relationship between the present and the immediate future.

@z 2 ịwðr;tị ỵ Vðr;tịwðr;tị ð3:3ị

When potential energy remains constant over time, solutions to the time-independent Schrödinger equation can be derived by expressing the wave function as w(r,t) = w(r)exp(iEt/ħ), where E represents the total energy of a particle with mass m.

Havingwðr;tịor wðrị, the average value of any physical quantityA(r) which depends only on the position vectorris given by the relation

Havingwðr;tịorwðrị, the average values of the momentumpor the squarep 2 of the momentum are given by the relations

Combining (3.5) and (3.6) with the non-relativistic expression for the total energyε,eẳ ðp 2 =2mị ỵ V, of a particle of massmweﬁnd

In writing (3.5)–(3.7) we assumed that ψ is normalized, i.e R drjwðrịj 2 ẳ1.Equations (3.3)–(3.7) were presented here for completeness; we shall try to avoid making use of them in the rest of this book.

Heisenberg ’ s Uncertainty Principle

Heisenberg's uncertainty principle asserts that the uncertainty in a particle's position, multiplied by the uncertainty in its momentum along the same direction, is always greater than or equal to h/2.

A waveparticle confined within a straight segment of length \( l_x \) cannot remain at rest; it must continuously oscillate This perpetual motion ensures that its average momentum squared, \( p_x^2 \), exceeds a certain threshold.

Heisenberg's principle is derived from the wave relation, which indicates that confining a wave packet within a spatial extent \( \Delta x \) necessitates the superposition of plane waves across a range of \( k \) values, with \( \Delta k \sim \frac{1}{\Delta x} \) By integrating this relationship with the wave-particle duality concept, we arrive at a fundamental understanding of quantum mechanics.

5Dxis the so-called standard deviation deﬁned by the relation:Dx 2 \ðx\x[ị 2 [ ẳ

\x 2 [\x[ 2 :A similar deﬁnition applies toDpxwithxreplaced bypx:The symbol , for any random quantity f, denotes its average value.

6 However, the average value ofpx can be zero, if the contributions of positive and negative values cancel each other.

3.2 The Properties of the Structures of the World 31 quantity of the order ofh 2 =l x 2 More precisely, conﬁnement of a waveparticle in a straight segment of lengthlx necessarily forces the waveparticle to go back and forth perpetually as to satisfy the inequality 7 (3.8), i.e.Dp 2 x ðh 2 Mx 2 ị:

For the usual three dimensional (3-D) case, where a particle may be conﬁned within a sphere of radiusRand volumeV ẳ ð4p=3ịR 3 , we obtain from (3.8) (and the choice\r[ ẳ\p[ ẳ0):

Combining (3.9) and (3.10) with the expression for the non-relativistic 9 kinetic energy, we obtain for the average value of the latter eKẳ \p 2 [

It's important to note that the numerical factors in (3.11) were derived under the assumption of a uniform probability density within the volume V Any variation in the probability density will generally alter these numerical factors Therefore, moving forward, we will typically exclude the numerical constants from our equations, leading to simplified relations.

In discussing the relationship between confinement length \( l_{x} \) and standard deviation \( \Delta x \), it is important to note that for a wave-particle confined in a fixed spatial region, one can set the coordinate system such that \( x = 0 \) Since this region is stationary, the momentum \( p_{x} \) is also zero Consequently, applying the general definition of standard deviation, we find that \( \Delta x^{2} = x^{2} \) and \( \Delta p^{2}_{x} \) holds true under these conditions.

In Quantum Mechanics, the relationship between the probability density and confinement length is crucial, as indicated by the equation \p 2 x [ h 2 =ð4\x 2 [ị For a uniform probability density, the relationship simplifies to \x 2 [ ẳDx 2 ẳl 2 x, highlighting that Dx is both proportional to and smaller than the confinement length This proportionality factor is influenced by the probability density, which is defined by the wave function w for the lowest energy state It is essential to recognize that the wave function remains as smooth as possible within the constraints of the potential, a concept that underpins the uncertainty principle As noted by Lieb, the uncertainty principle alone does not ensure the stability of structural formations in the universe.

8 Equation (3.10) was obtained by using the symmetry of the sphere and by assuming constant probability density for any value of r inside the sphere.

The correct relativistic relationship between kinetic energy (eK) and momentum (p) is expressed as eK = √(m₀²c⁴ + c²p²) - m₀c² In the non-relativistic limit, this simplifies to eK ≈ p²/(2m₀), while in the extreme relativistic limit, it approximates to eK ≈ cp Here, m₀ represents the rest mass of the particle This approach emphasizes the dependence on physical parameters and universal constants, allowing readers to focus on conceptual understanding rather than memorizing numerical factors In the problem section, readers may be tasked with restoring numerical factors and comparing their findings with the exact formulas.

Equation (3.11) is crucial as it highlights the essential non-zero kinetic energy required for stabilizing various structures in the universe It indicates that a wave-particle confined within a volume \( V \) possesses a minimum kinetic energy that cannot be zero This minimum energy is inversely related to the \( 2/3 \) power of the volume \( V \) and the mass of the wave-particle, while being directly proportional to the square of Planck’s constant \( h \) Consequently, a smaller volume and lighter mass result in a higher minimum average kinetic energy In classical physics, if \( h \) were zero, this minimum would also be zero, leaving the universe vulnerable to collapse due to attractive forces Importantly, the minimum kinetic energy approaches zero when \( V \) approaches infinity, indicating the wave-particle is no longer confined spatially.

In summary, (3.11) for the minimum average non-relativistic kinetic energy eKẳp 2 =2m of a particle of massm conﬁned within a ﬁnite volume V of radius

Rcan be rewritten (by omitting the numerical factor) as eK/ h 2 mV 2 = 3 / h 2 mR 2 ð3:13ị

In future chapters, we will explore how equations (3.11) and (3.13) are adjusted in the extreme relativistic limit, where energy approaches the speed of light, denoted as eK = c p This adjustment is crucial for understanding the implications of relativistic physics.

\p 2 [ p and (3.12), we obtain that the minimum kinetic energy of a particle conﬁned within a volume Vof radiusRin the extreme relativistic limit is: eK / ch

For a uniform probability distribution we have that\j jp [ ẳ0:968 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

The minimum kinetic energy of a wave-particle is inversely proportional to the one-third power of the confining volume or its radius, R This significant shift in the exponent of volume or radius is essential for understanding the transition from white dwarfs to neutron stars and the subsequent collapse of neutron stars into black holes.

Pauli ’ s Exclusion Principle

Identical wave particles sharing the same volume exhibit indistinguishability due to their wave-like motion, which eliminates any defined trajectory This indistinguishability implies that the physical properties of the particles remain unchanged when their "names" are interchanged The probability density, represented as the absolute square of the wavefunction Ψ, is a crucial physical property that describes the state of a system with multiple identical wave particles The indices associated with the wave particles indicate their respective positions and spin projections, further emphasizing their indistinguishable nature in quantum mechanics.

W(1,2, .ị j j 2 ẳjW(2,1, .ịj 2 , it follows that

In quantum mechanics, particles are categorized based on their spin: bosons, which have integer spin, and fermions, which have half-integer spin Bosons include particles like photons and gluons, while fermions encompass elementary particles such as electrons, neutrinos, and quarks, with protons and neutrons also classified as fermions due to their quark composition When two identical fermions occupy the same spatial and spin state, their joint wavefunction must change sign upon particle interchange, leading to the conclusion that the wavefunction equals zero This indicates that the probability of finding two identical fermions in the same state is zero, giving rise to the well-known Pauli exclusion principle.

Identical fermions, such as electrons, protons, neutrons, and quarks, adhere to the Pauli exclusion principle, which states that no more than one fermion can occupy the same spatial and spin single-particle state Specifically, for spin 1/2 fermions, a maximum of two can occupy the same spatial state—one with spin up and the other with spin down In simpler terms, this means that no more than two identical spin 1/2 fermions can exist in the same spatial region simultaneously.

Quantum Kinetic Energy in View of Heisenberg

When N spin-1/2 identical fermions are confined to occupy the same volume V, the analysis begins by determining the single-particle states defined by their momentum \( \hbar k \) and spin orientation, assuming no interactions These states are then organized by increasing energy, where the energy in the non-relativistic limit is given by \( E_k = \frac{\hbar^2 k^2}{2m} \) The N/2 lowest energy states are filled with two fermions of opposite spin, starting from the lowest energy level and progressing to the highest occupied energy, known as the Fermi energy \( E_F = \frac{\hbar^2 k_F^2}{2m} \) The Fermi momentum \( p_F = \hbar k_F \) is determined by ensuring that the sum of all momenta \( k \) from 0 to \( k_F \) (doubled for spin) equals the total number of fermions \( N \).

Having obtained the Fermi momentump F ẳhk F we can calculate the minimum total kinetic energyE K by summing over all occupied states:

Fig 3.1 The N/2 lowest energy single-particle spatial states are occupied by N spin ẵ fermions as to obtain the system ’ s ground state which is realized at T = 0 K

The summation over all occupied states appearing in (3.16) and (3.17) can be performed by the following general formula worth memorizing

Equation (3.18) can be generalized for systems of any dimension, as demonstrated in multiple-choice questions 3.8.9 and 3.8.10 The average energy per particle, denoted as e, is defined as EK/N, and it falls within the range of 0 and EF, as illustrated in Fig 3.1 This average energy represents the common kinetic energy of each fermion, which can be realized without violating Pauli’s exclusion principle by dividing the available volume V into N/2 equal subspaces, each with a volume of 2V/N By placing one pair of identical fermions, with one having spin up and the other spin down, in each subspace, we can derive the average kinetic energy per fermion using the non-relativistic expression in equation (3.13).

The average kinetic energy per particle for fermions confined in a reduced volume \( \frac{2V}{N} \) is expressed as \( e/h^2 m \left( \frac{2V}{N} \right)^2 = \frac{3}{h^2 N^2} = \frac{3 mV^2}{h^2 N^2} = \frac{3 mR^2}{h^2 N^2} \) Consequently, the total average kinetic energy of the \( N \) fermions, denoted as \( E_K \), is equal to \( N \) times the average kinetic energy per particle.

The proportionality factor in equation (3.20) can be derived from the observation that adding an additional fermion identical to the existing N fermions in volume V results in a minimum energy of EF for the new fermion, as dictated by Pauli's exclusion principle Consequently, the increase in the total minimum kinetic energy, represented as EK(N+1) - EK(N), equals EF This relationship implies that when dividing EK(N+1) - EK(N) by one and considering N to be a very large number, the proportionality factor can be established.

Equation (3.21) is applicable to systems of any dimensionality, including interacting systems, where the total energy Et replaces EK Special attention is needed when a spectral gap exists at the Fermi energy EF, particularly in semiconductors and insulators For odd N, the Fermi energy aligns with the highest occupied state, while for even N, it aligns with the lowest unoccupied state Consequently, Equation (3.21) calculates the mean value by positioning the Fermi energy at the center of the gap.

By combining (3.20) and (3.21) we obtain

The reader is encouraged to calculateEF by combining (3.16) and (3.18) Then by replacing it in (3.22a) he/she must show that in the non-relativistic limit the result is:

The equation EKẳ2:871h 2 N 5 = 3 mV 2 = 3 ẳ1:105h 2 N 5 = 3 m R 2 ð3:22bị aligns precisely with the result of (3.17) It's important to note that (3.19) indicates that, under conditions where identical fermions occupy the same space, the minimum average kinetic energy per fermion is enhanced compared to the Heisenberg-based (3.13) by a factor proportional to N² = 3! This enhancement can be significant when N is large, particularly in scenarios involving "free" electrons in metals.

The value of N 2 = 3 reaches approximately 10^15 per mole, which is a significant increase essential for stabilizing condensed matter at observed densities and compressibility levels, as will be discussed in an upcoming chapter.

In the extreme relativistic limit, the minimum total kinetic energy of spin-½ fermions occupying a spherical volume \( V \) with radius \( R \) can be determined using a method akin to the one previously described.

Schr ử dinger ’ s Principle of Spectral Discreteness

A classical wave confined in a finite space exhibits a discrete frequency spectrum, unlike the continuous spectrum of an infinite wave A prime example is a guitar string, which vibrates at specific frequencies—its fundamental frequency and harmonics—determined by the equation \( x_n = \frac{p t}{\lambda} \) This occurs due to the requirement for an integer number of half wavelengths to fit within the string's length Similarly, electromagnetic waves confined in a cavity also demonstrate a discrete spectrum Given the relationship \( E = hx \), we anticipate that a wave-particle confined in a finite region will exhibit similar discrete characteristics.

ﬁnite region of space, will acquire a discrete energy spectrum, in contrast to the continuous energy spectrum it would have according to the classical worldview.

In this book, we will link the concept of wave-attributable discreteness to Schrödinger, the pioneer of the equation that accurately describes the motion of non-relativistic wave-particles We can reformulate Schrödinger’s principle of discreteness in a clearer manner.

A wave-particle confined to a limited space possesses a specific set of discrete energy levels, denoted as e0, e1, e2, and so on These energy levels can be organized in ascending order, with no permissible energy values existing between any two consecutive discrete levels.

In the context of confined wave particles, their energy is characterized as "digital" rather than "analog." The lowest single-particle energy, denoted as \( e_0 \), exceeds the defined zero of energy by at least the minimum kinetic energy, as outlined in equation (3.13) Additionally, the energy difference \( \Delta e = e_1 - e_0 \) between the first excited state and the ground state is comparable in magnitude to \( e_0 \) in non-relativistic scenarios This relationship is represented by the equation \( \Delta e \approx \frac{c_1 \hbar^2}{mV^2} \approx \frac{c_2 \hbar^2}{mR^2} \), where \( c_1 \) and \( c_2 \) are numerical factors of order one, influenced by the potential type responsible for confinement Furthermore, the volume \( V \) in this equation may arise from either an external confining potential or internal interactions, as observed in composite wave particle systems.

Equation (3.24) holds significant importance, akin to the combination of Heisenberg’s and Pauli’s principles that culminate in (3.22b) This equation ensures the stability and existence of the structures within the Universe.

The schematic representation in Fig 3.2 illustrates the potential energy of a one-dimensional harmonic oscillator, defined as V(x) = 1/2 j x², along with its corresponding discrete energy levels Additionally, it presents the Coulomb potential energy, expressed as V(r) = e²/r, and the associated discrete energy levels.

The allowed values in the plane of total energy \(E_t\) versus frequency \(x\) exhibit distinct characteristics across different scenarios In the case of unconstrained classical waves, all points in the \(E_t\) versus \(x\) plane are permissible Conversely, confined classical waves are restricted to discrete lines perpendicular to the frequency axis, indicating frequency discretization For unconstrained particle-waves, only specific points on the lines defined by \(E_t = n h x\) (where \(n = 1, 2, 3, \ldots\)) are allowed, reflecting total energy quantization Confined particle-waves further limit permissible points to marked dots, showcasing both frequency discretization and total energy quantization In a similar vein, when considering total energy \(E_t\) versus particle energy \(\epsilon\), unconstrained non-interacting classical particles and wave-particles each exhibit specific allowed points along defined lines, with confinement introducing additional restrictions Ultimately, only cases involving wave-particle duality, specifically c, d, c', and d', accurately represent the quantum nature of reality, highlighting that classical and quantum behaviors converge as \(x\) or \(\epsilon\) approach zero.

3.6 Schr ử dinger ’ s Principle of Spectral Discreteness 39 implies that any temporary perturbation of size smaller thandewill leave a composite waveparticle unchanged in its ground state, corresponding to the minimum energyeo, if it was in this state initially This means that composite waveparticles, or even systems, behave as elementary ones up to a limit In other words, composite waveparticles, although they possess the possibility of being changed–even breaking–they require for this to happen the application of an external perturbation exceedinga non-zero energy value, which, in some cases, can be much larger than any available perturbation It is exactly this property which allows us to attribute a predictable chemical and physical behavior to, e.g., any atom of the periodic table, in spite of it being a composite system under the influence of quite different environments In contrast, in a classical world, wherehwould be zero, the behavior of an atom would never be the same, because the ever present interactions with its environment-even the most minute ones-would force the atoms to be in a situation of continuous change Equation (3.24) shows that the smaller a composite waveparticle is, the more stable it is For a nucleus, wheremẳmp1837me and

Non-radioactive nuclei remain stable under ordinary conditions, which is why alchemists were unable to transform common substances into gold using chemical energy, typically around a few electron volts (eV) This stability is attributed to the energy threshold required for nuclear reactions, which is significantly higher, around 40 MeV, making such transformations impossible with conventional methods.

The ground state energy and next excited energy level of wave-particles, such as electrons, and particle-waves, like photons, exhibit distinct differences due to the conservation of electron number versus the non-conservation of photon number This disparity allows for the consideration of the number of electrons in a system and its total energy as independent variables Consequently, the energy of a single electron in its ground state can be isolated, and any excitation of this one-electron system follows a linear progression, moving along a straight line or jumping to the next point in a sequence of energy levels.

In contrast to the depiction in Fig 3.3d′, the number of photons is not an independent variable; rather, it is determined by the total energy and, if applicable, the value of x Notably, the ground state energy (E_G), which represents the lowest possible energy of the electromagnetic field, corresponds to zero photons, yet it is not zero Instead, it is derived from the cumulative sum of all points along the line 0G, as illustrated in Fig 3.3c or d This phenomenon arises from the principles of Quantum Mechanics, which dictate that both the electric field (E) and the magnetic field are involved in this energy state.

The electromagnetic field cannot have both electric field (E) and magnetic field (B) equal to zero simultaneously, similar to how position (x) and momentum (px) cannot both be zero at the same time Consequently, the combined energy density, represented by E² + B², cannot be zero; its minimum value corresponds to the sum of all wave contributions per unit volume, denoted as 1/2 N_i hxi, where N_i indicates the number of waves at frequency xi This implies that the quantum vacuum of the electromagnetic field is not entirely devoid of activity, as it inherently contains fluctuations of the electromagnetic field vectors around their average values, highlighting the dynamic nature of what may seem like empty space.

The lowest excited energy level of the electromagnetic field occurs when a system transitions vertically from point 0G to point 01, resulting in an energy state of EG + ħx, which corresponds to a single photon of frequency x Higher frequencies require greater minimum excitation energy, indicating that in the quantum realm, increased frequency necessitates more external energy for excitation compared to classical systems The most effective method for exciting a system from its ground state is by introducing it to a heat bath at absolute temperature T, allowing energy transfer to the wave-particle within a range from 0 to approximately kBT, where kB represents Boltzmann’s constant.

A heat bath at temperature \( T \) in equilibrium with a photon system cannot significantly excite photons with frequencies exceeding \( x_M = c_1 \frac{k_B T}{h} \), where \( c_1 \) is a constant of order one This quantum mechanical limit is crucial for calculating the total electromagnetic energy \( I \) emitted by a black body per unit time and area The energy \( I \) can be derived by integrating the contribution \( I(x) \, dx \) over the frequency range \( [x, x + dx] \) Classically, \( I(x) \, dx \) is proportional to \( \left( \frac{k_B T}{c^2} \right) x^2 \), but integrating this from zero to infinity yields an unphysical result By restricting the upper limit of integration to the quantum mechanical maximum frequency \( x_M \), the result becomes proportional to \( \frac{(c_1^3/3)(k_B T)^4}{c^2 h^3} \), aligning with experimental data.

In conclusion, R Feynman's depiction of wave-particle propagation illustrates that, unlike a classical particle which follows a singular path from point A to point B, a wave-particle explores all possible trajectories between these points, each assigned a probability amplitude The probability of the wave-particle traveling from A to B is determined by the absolute value squared of the sum of these amplitudes Typically, significant contributions to this sum arise from trajectories confined within a cross-sectional area proportional to the square of the wavelength around the classical path This path integral formulation of quantum mechanics is equivalent to the traditional approach governed by Schrödinger’s equation, offering a more intuitive understanding of particle behavior.

10 For the numerical values of all universal constants see Table I.1 in Appendix I.

Summary of Important Concepts and Formulae

The uncertainty principle indicates that a wave-particle confined within a limited space must have a non-zero kinetic energy, which increases as the confinement length decreases This characteristic is crucial as it facilitates the establishment of equilibrium in composite systems As attractive interactions compress the system, the kinetic energy rises until it balances the squeezing pressure from these interactions, leading to a state of equilibrium.

The uncertainty principle asserts that the product of the standard deviations of position and momentum components, such as Dx and Dp x, is always greater than or equal to h/2.

Standard deviations are defined by the relationships \( D_x^2 \leq \langle x^2 \rangle \) and \( D_p^2 \leq \langle p^2 \rangle \) For a wave-particle confined in a fixed region of space, it is possible to select a coordinate system where \( \langle x \rangle = 0 \) and \( \langle p \rangle = 0 \).

Dx 2 ẳ \x 2 [ and Dp 2 x ẳ\p 2 x [ It follows that \p 2 x [ h 2 =ð4\x 2 [ị

; thus the minimum value of\p 2 [ is

From (3.9) it follows that the minimum averagenon-relativistickinetic energy eKẳp 2 =2m of a waveparticle of mass m conﬁned within a ﬁnite volume V is necessarily non-zero and proportional to eK / h 2 mV 2 = 3 / h 2 m R 2 ð3:13ị

Also from the relationeK ẳcj jp, valid in the extreme relativistic limit, and the relation \j jp [ / ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

\p 2 [ p we have that the minimum average extreme relativistickinetic energyeKẳcj jp of a particle conﬁned within aﬁnite volumeVis necessarily non-zero and proportional to eK/ h c

The exclusion principle applies exclusively to identical fermions occupying the same volume \( V \) When \( N \) fermions, each with spin \( \frac{1}{2} \), are divided into \( N/2 \) equal subspaces of volume \( \frac{2V}{N} \), placing one fermion with spin up and another with spin down in each subspace adheres to the exclusion principle However, this division increases the minimum average kinetic energy of each fermion by a factor of \( N^2 = 3 \) and \( N^1 = 3 \) in specific cases Consequently, the total minimum kinetic energy \( E_K \) of \( N \) identical spin \( \frac{1}{2} \) fermions in the same volume is significantly affected.

E K /h 2 N 5 = 3 mV 2 = 3 /h 2 N 5 = 3 m R 2 ð3:22ị while for theextreme relativisticlimit is

The energy difference (de) between the first excited state and the ground state of a single particle confined in a finite volume (V) and radius (R) is described by specific formulas In the non-relativistic limit, it is expressed as de = (c1 * h²) / (m * V²) = (c2 * h²) / (m * R²) In contrast, the extreme relativistic limit simplifies this relationship to de = (c1 * h * c).

3.7 Summary of Important Concepts and Formulae 43

1 An isolated composite system left undisturbed would eventually end up in a final state which is determined by one of the following mechanisms:

The fundamental forces acting between particles in a system exhibit attractive behavior when the distance between them (d) exceeds a specific threshold (a), while they become repulsive when the distance is less than this threshold Consequently, the system achieves an equilibrium state at the distance where d equals a A prime example of this phenomenon can be observed in diatomic molecules.

The final state of a system is influenced by initial conditions and various random processes, making it not uniquely determined A prime example of this concept is the orbit of a planet around the Sun.

The equilibrium state of a system is influenced by two opposing factors: the attractive potential energy from interactions, which can cause the system to collapse, and the total kinetic energy relative to the center of mass, which tends to disperse the system A prime example of this balance is found in the hydrogen atom.

Every system is bound to collapse over time due to the ongoing emission of radiation and the gradual loss of mechanical energy A prime illustration of this phenomenon can be observed in artificial satellites orbiting Earth.

For each of the above four answers/statements argue against or in favor.

2 The minimum kinetic energy of a system of N identical spin ẵ particles (of zero total momentum and angular momentum) confined within a finite 3-D space of volume

V= 3 π R is given by one of the following formulas (if ε K =p 2 / 2m ):

3 The minimum kinetic energy of a system of N identical spin ẵ particles (of zero total momentum and angular momentum) confined within a finite 3-D space of volume

V= 3 π R is given by one of the following formulas (if ε = K c p ):

In a one-dimensional harmonic oscillator, the standard deviation \( x \) represents the deviation from the classical equilibrium position in the ground state, characterized by mass \( m \) The energy difference between the ground state and the first excited state can be expressed using specific formulas.

5 In the ground state of a hydrogen atom, the electron is confined (with a probability of about 76%) within a spherical volume of radius r = 2 2 /m e e 2 around the proton The energy difference between the first excited and the ground state is given by one of the following formulas:

6 The relation between the average kinetic energy per particle ε≡E K /N of N identical spin ẵ particles of zero total momentum and angular momentum confined within a finite 3-D space of volume V= 4 3 π R 3 and the corresponding Fermi energy is given by one of the following formulas (assume that ε = K p 2 / 2 )m :

7 The relation between the average kinetic energy per particleε ≡E K /N of N identical spin ẵ particles of zero total momentum and angular momentum confined within a finite 3-D space of volume V= 4 3 π R 3 and the corresponding Fermi energy is given by one of the following formulas (assume thatε = K c p ):

8 The relation between the average kinetic energy per particle ε≡E K /N of N identical spin ẵ particles of zero total momentum and angular momentum confined within a finite 2-D (two-dimensional) space of area A= πR 2 and the corresponding Fermi energy is given by one of the following formulas (assume thatε = K p 2 / 2m ):

9 The Fermi energy of N identical spin ẵ particles of zero total momentum and angular momentum confined within a finite 2-D (two-dimensional) space of area A= πR 2 is given by one of the following formulae (assume thatε = K p 2 / 2m ):

(For the 2-D case we have (2 ) A 2 d k 2

10 The relation between the average kinetic energy per particle ε≡E K /N of N identical spin ẵ particles of zero total momentum confined within a finite 1-D (one- dimensional) space of length L=2a and the corresponding Fermi energy is given by one of the following formulas (assume thatε = K p 2 / 2m ):

(For the 1-D case we have 2 k

The correct answers to the multiple choice questions are given at the end of the book (see Appendix H).

Solved Problems

1 By employing Heisenberg’s uncertainty principle, obtain the minimum average total energy of a one-dimensional harmonic oscillator in terms of the uncertainty Dx in its position The oscillator is characterized by its ‘spring’ constant j and its mass m Minimize with respect to Dx in order to determineDx and.

In the context of a harmonic oscillator, the displacement \( x \) oscillates symmetrically around the classical equilibrium position, which is defined as zero Consequently, the mean displacement \( \langle x \rangle \) equals zero, and the mean square displacement \( D x^2 \) can be expressed as \( \langle x^2 \rangle \) Similarly, the momentum variance \( D p^2 \) is derived from the same principles, leading to the conclusion that the energy \( \langle E \rangle \) is given by \( \langle p^2 \rangle / (2m) \).

In this analysis, we apply Heisenberg's uncertainty principle, represented as DxDph = 2, to minimize the expression (h²/8mDx²) + (jDx²/2) By taking the derivative with respect to Dx and setting it to zero, we derive that Dx² equals h/(2√(jmₚ)) Consequently, we find that (h²/8mDx²) equals (jDx²/2), leading to the conclusion that hx equals 4 and thus, x is √(j/mₚ) This yields an exact result consistent with quantum mechanics principles.

\e[ ẳhx=2 and the average kinetic energy is equal to the average potential energy.

2 * 11 Consider a one-dimensional shallowðV ðh 2 =m a 2 ị eịpotential energy well Vðxị of depth –V and extent 2a For xj j[a; Vðxị ẳ0: Can such a potential well bound a particle of mass m?

In a bound state, a particle is confined and cannot escape to infinity, which implies its energy must be lower than the potential energy at infinity, set at zero The extent of confinement, denoted as L, must be significantly larger than the characteristic length a; otherwise, the quantum kinetic energy would result in a positive total energy, contradicting the bound state condition When L is much greater than a, the potential energy can be approximated as V(a/L), where V is the potential energy when the particle is within the well, multiplied by the probability of the particle being found in that region To find the optimal confinement length, one must minimize the sum of the kinetic and potential average energies with respect to L.

L of the conﬁnement, we ﬁnd: Lẳ ð1=c P ịðh 2 =ma 2 Vịaẳ ð1=c P ịðe=Vịa and

In the limit as V approaches zero, the exact results indicate that for cP equal to 2, the relationship L equals a times e divided by 2V and E equals 2V squared divided by e holds true This implies that there is always at least one bound state present, regardless of how shallow the one-dimensional potential well may be Furthermore, the binding energy for very shallow one-dimensional potential wells is expressed as the ratio V squared divided by h squared, multiplied by 2m a squared.

11 Problems indicated by an asterisk have broader physical implications, but require more than a simple application of a formula.

3 *Consider a three-dimensional potential well (Vðrị ẳ V; for r\a and

Vðrị ẳ0; forr[aị Under which conditions can such a potential well sustain at least one bound state for a waveparticle of mass m?

SolutionAs we shall show, in three dimensions (in contrast to what happens in one-dimension) a very shallow potential well cannot sustain a bound state; the ratioV=h 2 =m a 2

For at least one bound state to exist, the potential V must exceed a critical value This requirement arises from the fact that in three dimensions, the probability density acquires an additional factor of 1/r² compared to one dimension This can be illustrated by considering a spherically symmetric electromagnetic wave, where energy conservation dictates that the energy flux through any spherical surface remains constant, leading to the conclusion that the electromagnetic energy density is inversely proportional to the square of the radius A similar principle applies to probability density, necessitating the presence of the 1/r² factor This factor, combined with the continuity of the wave function and its radial derivative, indicates that the assumption of negligible kinetic energy in the region r < a (as used in one-dimensional cases) is invalid in three dimensions Instead, the kinetic energy contribution from the region r < a is proportional to the probability of finding the wave particle within the potential well By considering both the kinetic energy from the regions r < a and r > a, along with the potential energy, we derive the total energy E Minimizing this energy with respect to L leads to specific relationships between the parameters, ultimately yielding the critical value of V.

In a neutral atom with a large atomic number Z, the total energy is influenced by the mean distance r between an electron and the nucleus, the average distance a between two electrons, and the atomic radius R The total energy comprises three components: the quantum kinetic energy of the electrons, the Coulomb attractive potential energy, which is proportional to Z²e²/r, and the repulsive mutual potential energy among the Z electrons, represented as Z(Z-1)e²/a To express the total energy of the atom in terms of R, the ratios r/R (denoted as x) and a/R (denoted as y) must be utilized.

For optimal convenience, it is advisable to operate in atomic units and convert back to conventional units at the conclusion of the calculations The total kinetic energy can be expressed as \( E_K = 1.105Z^5 = 3/R^2 \) Additionally, the Coulomb interactions between electrons play a significant role in the overall energy dynamics.

3.9 Solved Problems 47 nucleus isZ 2 =rẳ Z 2 =x Rand the Coulomb repulsions among electrons is

The total energy equation is represented as E = 1.105(Z^5 / R^2) b(Z^2 / R), where b is a function of (1/x)(1/2y) By minimizing the total energy, we set the derivative with respect to R equal to zero, while keeping x and y constant, to derive the optimal value for R.

Rẳ ð2:21=bịZ 1 = 3 Substituting this value ofRinEweﬁnd that

In their book "Quantum Mechanics, 3rd ed.," L.D Landau and E.M Lifshitz address this issue using the Thomas-Fermi approximation, an advanced technique that not only confirms our simpler method's results but also reveals the distribution of electrons surrounding the atom Both methods yield the same dependence on the underlying principles.

The sophisticated Thomas-Fermi approach applies to both white dwarfs and neutron stars, where the exponents (−1/3) and (7/3) are relevant By selecting b = 1.61, which aligns with reasonable values of ra0:31R, we can derive the experimental value of E When restoring ordinary units with b = 1.61, we find R = 0.73Z^(1/3) A.

The value of E 16Z 7 = 3 eV indicates a small radius (R) that defines a spherical surface where most electrons are located This radius is distinct from the atomic radius determined by the highest occupied orbital.

Unsolved Problems

4 For a one-dimensional attractive potential of the formVðxị / j jx b , whereβis a constant, show that the energy of the nth state is

For the potential well of inﬁnite depth we have b! 1; and aẳ2; for a harmonic oscillator bẳ2; and aẳ1; and for a Coulomb-like potential bẳ 1; andaẳ 2.

(HintðdE=dnịdn/ ðdV=dxịk; and k/E 1 = 2 ).

5 Consider a non-relativistic particle of massmconﬁned within an ellipsoid with main semi axesa,b, andc What is its minimum kinetic energy? How does this kinetic energy compare with what would result if the ellipsoid becomes a sphere of equal volume? The probability density toﬁnd the particle within the ellipsoid or the sphere is assumed constant.

6 *Consider a two-dimensional shallow ðV ðh 2 =m a 2 ị eị potential energy well of depth –V and radius a Is such a potential well capable of binding a particle of massm?

Kinetic energy is proportional to \( h^2 = 2mL^2 \), while potential energy is related to \( a^2 = L^2 \ln(L/L_0) \), with the logarithmic term arising from the wave function's Bessel function characteristics (see Economou [9]) The two-dimensional case serves as a transitional point between one-dimensional and three-dimensional scenarios regarding bound states in shallow potential wells Understanding bound states across 1D, 2D, and 3D potential wells has significant implications in various physical applications.

Low-frequency sound waves in solids exhibit a relationship similar to electromagnetic (EM) waves in a vacuum, where their frequency is proportional to the wavevector k At low temperatures, the average energy of thermal excitations from sound waves in a solid is proportional to the fourth power of temperature, leading to a specific heat that is proportional to the third power of temperature In classical physics, this relationship differs, highlighting the unique behavior of sound wave energy and specific heat in solids at low temperatures.

Problems indicated by an asterisk have broader physical implications, but require more than a simple application of a relevant formula.

1 R Feynman, R Leighton, M Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, 1964)

2 A Einstein, B Podolsky, N Rosen, Phys Rev 47, 777 (1935)

3 A Aspect, P Grangier, G Roger, Phys Rev Lett 49, 91 (1982)

4 R Grif ﬁ ths, Consistent Quantum Theory (Gambridge University Press, Cambridge, 2002)

5 P Ghose, Testing Quantum Mechanics on New Ground (Gambridge University Press, Cambridge, 1999)

6 A Peres, Quantum Theory: Concepts and Methods (Kluwer Academic Publishers, Dordrecht, 1995)

7 H Paul, Introduction to Quantum Theory (Gambridge University Press, Cambridge, 2008)

8 R Shankar, Principles of Quantum Mechanics, 2nd edn (Springer, NY, 1994)

9 E.N Economou, Green ’ s Functions in Quantum Physics, 3rd edn (Springer, Berlin, 2006)

10 L.D Landau, E.M Lifshitz, Quantum Mechanics, 3rd edn (Pergamon Press, Oxford 1980) (Butterworth-Heinemann, Great Britain, 3rd rev and enl edn 2003)

Equilibrium and Minimization of Total

The combination of the First and Second Laws of Thermodynamics, along with the relationship dW = P dV, leads to the equation dU = T dS - P dV + μ dN This equation highlights that the chemical potential (μ) plays a crucial role in the thermodynamic process The inequality derived from this equation is valid during the approach to equilibrium, where energy (U) decreases and reaches its minimum value at equilibrium By adjusting the independent variables from entropy (S) to temperature (T), volume (V) to pressure (P), and particle number (N) to chemical potential (μ), we can derive various thermodynamic potentials, including the Gibbs free energy (G).

UþPVTSsatisfying the relationdG SdTþVdPþldN which means that,under conditions of constant temperature and pressure and no exchange of matter,equilibrium corresponds to the minimum ofG.

Concepts

Equation (4.1) indicates that a physical system at negligible absolute temperature, when isolated from external work and m-particles, achieves a stable equilibrium at the minimum of its total energy, U.

Internal energy (U) is defined as the average total energy (E_t) of a system when the total momentum is zero For macroscopic systems, this also requires zero average total angular momentum The values of U and E_t are determined relative to a chosen reference state, typically where all particles are infinitely distant and in their ground state U or E_t comprises three contributions: the relativistic rest energy (E_0 = m_0c^2), kinetic energy (E_K), and potential energy (E_P), which includes interactions with the environment and among particles Often, the rest energy is incorporated into the reference state if it remains constant.

In a physical system interacting with an environment at constant temperature (To) and constant pressure (Po), the stable equilibrium state of the system is achieved when the availability, a key thermodynamic quantity, reaches its minimum value.

The equation AUỵPoVToS ð4:2ị relates the total energy (U), volume (V), and entropy (S) of a system, highlighting the principles of minimization derived from the second law of thermodynamics, which states that entropy tends to increase This relationship can be further explored by expanding UðV;Sị in powers of V.

In the context of thermodynamics, the minimization of availability A leads to the conclusions that pressure P equals the reference pressure Po and temperature T equals the reference temperature To, alongside certain general thermodynamic inequalities Consequently, at equilibrium, the availability A aligns with the Gibbs free energy, expressed as GU + PVTS, which will be further explored in Section 4.4.

This chapter introduces the first and second laws of thermodynamics, highlighting their essential role in understanding equilibrium structures Additionally, it presents other significant thermodynamic relations For further details, refer to [2].

Conservation of Energy and the First Law

The first law of thermodynamics, known as the law of conservation of energy, asserts that the infinitesimal change in the internal energy (dU) of a system results from three interactions with the environment: (a) the absorption of an infinitesimal amount of heat (dQ), (b) the work done by the system on the environment (dW), and (c) the influx of an infinitesimal quantity of energy (dEm) associated with the entry of m-particles Consequently, the principle of energy conservation can be expressed as dU = dQ - dW + dEm.

The quantities dQ, dW, and dEm are not perfect differentials of specific functions Q, W, and Em, indicating that such functions do not exist The internal energy of a system cannot be divided into these three components, as this separation only holds meaning during the exchange or transformation of energy Mathematically, dQ, dW, and dEm are classified as differential forms rather than perfect differentials, a distinction represented by the symbol 'd' instead of 'd'.

2 These inequalities are:T0; CV[0; CP[CV; ð@P=@Vị T \0 [3].

52 4 Equilibrium and Minimization of Total Energy

Entropy and the Second Law

The second law of thermodynamics, a fundamental principle of nature, connects macroscopic phenomena to their microscopic origins through the concept of entropy Entropy is defined in systems with numerous microscopic particles, making detailed descriptions impractical; thus, a macroscopic overview becomes essential It is proportional to the logarithm of the total number of microscopic states corresponding to a specific macroscopic state, which is characterized by independent variables such as energy (U), volume (V), and particle number (N).

SkBlnCðU;V;N; .ị; ð4:4ị k B ẳ1:3810 23 m 2 kg s 2 K 1 is a universal physical constant called Boltzmann’s constant.

The second law of thermodynamics states that the entropy of an isolated system, which does not exchange heat or particles with its surroundings, consistently increases until it reaches maximum entropy at equilibrium This principle indicates that processes occurring within the system as it approaches equilibrium will enhance its entropy, while any reverse processes that could potentially reduce entropy are deemed impossible without external intervention, thus violating the second law Consequently, the journey toward equilibrium in an isolated system is characterized by irreversible processes, such as heat transfer between components at varying temperatures, diffusion of particles with differing chemical potentials, and internal chemical reactions Only in rare extreme cases can such processes be considered reversible, provided they do not alter the entropy.

When a system exchanges heat with the environment without the transfer of m-particles, entropy changes in two key ways: internally, through irreversible processes occurring within the system, and externally, as a result of the heat exchange with the surroundings.

3 A simple system, such as a perfect gas, when in equilibrium, can be described macroscopically by only three independent macroscopic variables, e.g.U;V;N;for a photon system in equilibrium

Nis is not an independent variable, indicating that only two independent variables are necessary for its macroscopic description In contrast, more complex equilibrium systems may need additional independent macroscopic variables Furthermore, non-equilibrium macroscopic states require a greater number of independent variables compared to their equilibrium counterparts The exchange of heat affects the system's entropy, while the exchange of work does not.

The change in entropy of an open system that exchanges heat with its environment can be positive, negative, or zero, depending on the conditions According to the second law of thermodynamics and the positivity of temperature at equilibrium, this relationship can be further understood through an alternative formulation.

Combining (4.3) and (4.6), i.e., the 1st and the 2nd law, we obtain: d UT d Sd Wỵd Em ð4:7ị

Equation (4.7) indicates that when TdS equals dW equals dEm equals 0, the internal energy (U) of the system continues to decrease until it reaches an equilibrium state where energy is minimized Specifically, equation (4.8) suggests that if the internal energy (U) at equilibrium depends solely on entropy (S), volume (V), and the number of particles (N), then U will still decrease on the path to equilibrium, even when S, V, and N remain constant.

Inequality (4.7) as a Source of Thermodynamic Relations

In this section we will make some remarks which may help the reader in manipulating thermodynamic inequalities or equalities and therefore reducing the need for memorization.

To utilize the relationship (4.7), we require explicit expressions for dW and dEm The well-known formula for dW is PdV, which represents the mechanical work performed by gases and liquids It's important to note that other forms of work exist, such as the electrical work expressed as /dq, where / denotes electrostatic potential and dq signifies the change in the system's charge In a similar manner, the infinitesimal quantity dEm can be represented as ldN, where μ is the chemical potential, to be elaborated on in the following section By substituting these explicit expressions into (4.7), we derive the equation dU = TdS - PdV + ldN.

In two scenarios, equation (4.8) simplifies to equality: first, when the system remains in equilibrium or experiences minimal internal entropy changes; second, when the function U(S, V, N) is derived under the assumption of thermodynamic equilibrium, where dU represents the differential of this function This leads to the relationships dU = (∂U/∂S)V,N dS + (∂U/∂V)S,N dV + (∂U/∂N)S,V dN By comparing this with dU = TdS - PdV - ldN, we derive that (∂U/∂S)V,N = T, (∂U/∂V)S,N = -P, and (∂U/∂N)S,V = -l Further useful relations can be obtained by differentiating equation (4.9), such as (∂T/∂V)N = (∂²U/∂S∂V)N = (∂²U/∂V∂S)N = (∂P/∂S)N.

In equation (4.8), the upper bound on the quantity dU includes the differentials of the independent variables S, V, and N, which we define as the natural independent variables of U However, from an experimental perspective, it may be more practical to control a different set of independent variables, such as T, V, and N, instead of S, V, and N To facilitate this transition from S to T, we can manipulate the equation by adding or subtracting the product TS, where the differential of TS is expressed as d(TS) = TdS + SdT Consequently, we can derive the differential d(UTS) as TdS - SdT + PdV - ldN To eliminate the term TdS, we subtract TS from U, leading us to define the new quantity F as U - TS, which simplifies the differential to dF = d(UTS) = -SdT + PdV - ldN.

Helmholtz free energy is defined with natural independent variables T, V, and N The relationship between these variables holds true under the same conditions as previously established By manipulating natural independent variables through the addition or subtraction of thermodynamic quantities, one can transition between different sets of variables For instance, it is possible to shift from the set S, V, N to T, P, N—commonly used in experimental settings—by incorporating the term TS + PV into the internal energy U.

F the quantity PV The quantity GUTSþPV FþPV is the so-called Gibbs free energyintroduced in Sect.4.1; its differential satisﬁes the relation dG SdT ỵVdPỵd E m ð4:12ị

Under constant temperature and pressure, with no exchange of m-particles, the Gibbs free energy (G) decreases as a system approaches equilibrium, as indicated by the equation (4.12) This behavior is consistent with normal experimental conditions, where G reaches its minimum value at equilibrium, with only extreme cases allowing for it to remain unchanged.

Maximum Work, Gibbs ’ Free Energy, and Chemical

Let us consider now a systemnot in equilibriumand in contact with an environment of constant temperatureToand constant pressurePo What is the maximum work,

Work can be extracted from a system in a non-equilibrium state, with the maximum work achieved when the total entropy of the system and its environment remains constant throughout the process This condition leads to a specific formula for calculating the maximum work, denoted as W M.

WMẳAinitialAfinal ð4:13ị whereAis the availability deﬁned in (4.2) (See Unsolved Problem 3) IfPẳPo and T ẳTo then the maximum work becomes (taking into account (4.2) and

Equations (4.13) and (4.14) provide a justiﬁcation for the names availability and free energy forAand Grespectively.

As an elementary application of (4.7), let us minimizeUẳE K ỵE P with respect to the volumeVof the system, under the conditionsT o ẳP o ẳd E m ẳ0 We have ð@U=@Vị ẳ ð@EK=@Vị ỵ ð@EP=@Vị ẳ0 ð4:15ị

In equilibrium processes, where the system conditions are defined by ToẳPoẳd Emẳ0, the relationships ð@U=@Vị ẳ P, ð@EK=@Vị ẳ PK, and ð@EP=@Vị ẳ PP hold true Consequently, minimizing the energy U concerning volume under these conditions leads to the significant conclusion that, at equilibrium, PK equals PP This indicates that the expanding pressure of the system is balanced.

56 4 Equilibrium and Minimization of Total Energy kinetic energy equals the absolute value of the squeezing pressure of the potential energy.

For processes transitioning from one equilibrium state to another while maintaining a constant entropy (dS = 0), the equality holds true in equations (4.7), (4.8), (4.11), and (4.12) This relationship is further illustrated by considering equation (4.12).

T ; P ẳl ð4:16ị where the subscripts indicate which variables are kept constant during the differ- entiation By differentiating once more (4.16) we can obtain further thermodynamic relations in analogy with (4.10).

The chemical potential (μ) is a crucial quantity in thermodynamics, as it must remain constant throughout a system at equilibrium, similar to temperature In non-equilibrium states, particles migrate from areas of high μ to low μ until equilibrium is restored Additionally, μ is directly related to Gibbs' free energy For photons and other non-conserved particles, the chemical potential is zero, as their number is not an independent variable, eliminating the need for the term dE/dN in relevant equations.

Extensive and Intensive Thermodynamic Quantities

Most thermodynamic systems are influenced by short-range forces, particularly in macroscopic systems at equilibrium, where electromagnetic forces play a crucial role For these systems to maintain equilibrium, they must remain locally electrically neutral, causing the effects of positive and negative charges to cancel out and resulting in short-range interactions on the order of Angstroms However, there are exceptions where long-range forces, such as gravitational forces, become significant, making traditional thermodynamic relations inapplicable on a global scale These long-range forces will be explored further in Part V of this book, along with the significant role of long-range interactions in atomic nuclei, which will be discussed in Chapter 9.

In thermodynamic systems with short-range forces, properties can be categorized as either proportional to the number of particles or independent of it Internal energy, entropy, volume, and Gibbs free energy are examples of extensive properties that depend on particle count, while pressure, temperature, and chemical potential are intensive properties that do not This distinction allows us to analyze the Gibbs free energy's dependence on temperature, pressure, and particle number Since Gibbs free energy is extensive, it is proportional to the number of particles (N), with the proportionality coefficient being a function of the intensive variables temperature (T) and pressure (P), along with universal constants that characterize the particles Thus, Gibbs free energy can be expressed as G = Nf(T, P, {u.c.}) By applying the relevant equations, we establish that f(T, P, {u.c.}) = l(T, P, {u.c.}).

In future sections, we will explore the intensive and extensive properties of thermodynamic quantities, particularly in the context of questions and problems, as well as in later chapters involving dimensional analysis It is important to note that the combinations N₅ = 3 = V₂ = 3 and N₄ = 3 = V₁ = 3, which represent the total kinetic energy of systems composed of identical fermions, are classified as extensive quantities.

Summary of Important Relations

The conservation of energy as expressed by the 1st Law, d Uẳd Qd Wỵd Em ð4:3ị in combination with the deﬁnition of entropy,

Sk B lnCðU;V;N; .ị; ð4:4ị and the 2nd Law in the form

T d Sd Q; ð4:6ị leads to the basic relation d UT d Sd Wỵd Em ð4:7ị

58 4 Equilibrium and Minimization of Total Energy which, afterd Wandd Emare replaced by explicit expressions, is the starting point for the direct derivation of a large number of thermodynamic relations.

In the simple and usual case ofd WẳPdV andd EmẳldN; (4.7) becomes d UT d SPdVỵldN ð4:8ị

Starting from the set of natural independent variables S; V; N related to internal energy U, we can derive any combination of three independent variables By strategically adding or subtracting products of suitable thermodynamic quantities, we can achieve sets such as T; V; N, T; P; N, or T; V; l.

FUTS; or GFþPV UTSþPV; or XFlNUTSlN respectively: dF SdTPdVỵldN ð4:17ị dG SdT ỵVdPỵldN ð4:18ị dX SdTPdVNdl ð4:19ị

Equation (4.19) with its set of natural independent variables T; V; l is introduced because certain second derivatives such as @ 2 X=@T 2

V ;lẳ ð@S=@Tị V :l are much easier to calculate, at least in the case of non-interacting fermions.

If there is no change in internal entropy, the equations (4.8), (4.17), (4.18), and (4.19) become equalities In this scenario, applying the first and second derivatives to these equalities generates six additional thermodynamic relations, similar to those found in (4.9) and (4.10).

In this book, we highlight a crucial point: we have developed a method to predict or justify the equilibrium structures of matter and their properties.

To determine the Gibbs free energy (G) or internal energy (U) for the system under study, we analyze conditions of constant pressure and temperature (P and T) with no change in energy (dE = 0) This involves expressing G or U as a function of various free parameters, including volume, atomic or ionic positions, and electronic concentration (ne) By minimizing G or U concerning these parameters, we identify a state of stable equilibrium that can be observed in nature.

1 Which one of the following schematic graphs of S vs U (under constant V and N) is consistent with physical reality?

2 Which one of the following schematic graphs of G vs T (under constant P and N) is consistent with physical reality?

3 Which one of the following schematic graphs of G vs P (under constant T and N) is consistent with physical reality?

4 Which form of the dependence of U on S V N , , is explicitly consistent with the intensive/extensive character of thermodynamic quantities?

5 Which form of the dependence of F on , T V N , is explicitly consistent with the intensive/extensive character of thermodynamic quantities?

6 Which form of the dependence of Ω on T V , , μ is explicitly consistent with the intensive/extensive character of thermodynamic quantities?

7 In the graphs below, the entropy vs internal energy for a thermaly isolated system with m 0

E = is shown together with a point P representing an initial non-equilibrium state Which one is consistent with physical reality?

8 In the graphs of question 7 indicate the maximum work which can be obtained by exploiting the non-equilibrium initial state ( W max = U i − U f )

9 In the graphs below, the Gibbs free energy vs temperature (under constant pressure) is plotted for each of the three phases of matter (solid,S, liquid,L, gas,G) Taking into account the definition of entropy and the properties of G, indicate which graph is consistent with physical reality.

10 As equilibrium is approached under conditions of being kept constant, the energy U is decreasing This decrease to be consistent with energy conservation must be accompanied by one of the following processes:

(a) Work is done by the system (b) Outflow of mass is taking place

(c) Outflow of heat occurs (d) The system lowers its pressure

Solved Problems

1 Using the intensive/extensive feature and (4.19) prove thatXẳ PV

In considering the natural independent variables for Ω, which are T, V, and l, we recognize that among these, only V is an extensive variable Therefore, it follows that the extensive quantity Ω must be expressed in the form Xẳ.

VfðT;lị: Taking into account this last relation and (4.19) we have that f(T;lị ẳ ð@X=@Vị T ;l ẳ P QED.

2 A homogeneous and isotropic material of volume V and of magnetic suscepti- bilityχ(where by deﬁnition M =χH with M being the magnetization and H the auxiliary magneticﬁeld) is placed in a uniform externally controlled magnetic

ﬁeld H It is given thatvẳA=T; where A is a positive constant and T is the temperature, and that d W ẳ VHdM:

Calculate (a) the entropy S(H, T) in terms of S(0, T) and in terms of the other relevant parameters, (b) the derivative ð@T=@Hị S : Is there any physical application of this quantity?

Solution Equation (4.7), assuming no transfer of matter (d E m ẳ0) and no production of internal entropy, becomes in the present case, dUẳTdSỵVHdM ð4:20ị

Both SandMare are challenging independent variables due to their difficulty in control and limited relevance to the desired outcomes In contrast, the more straightforward independent variables are temperature (T) and magnetic field (H) A common method for transitioning from S and M to T and H involves adjusting the energy (U) by subtracting the product of temperature (T) and entropy (S) along with the product of volume (V) and magnetic field (H).

Vis constant) dJẳ SdTVMdH ð4:21ị from which it follows by taking second derivatives as in (4.10) that ð@S=@Hị T ẳVð@M=@Tị H ẳVHð@v=@Tị ẳ VHA=T 2 ð4:22ị

Integrating (4.22) with respect toHfromH= 0 to itsﬁnal value underT= const. and taking into account thatA=T 2 ẳv=T, we have

The differential of (4.23) is dSẳẵdSð0;Tị=dT dTỵ(VH 2 v=T 2 ịdT(VHv=TịdH ð4:24ị SettingdS= 0 in (4.5) we obtain the desired derivative ðdT=dHị S ẳ VHv

In (4.24) we made use of the relationCV ẳ ð@U=@Tị V ẳTð@S=@Tị V Notice that the derivative in (4.25) is positive Thus by reducing the magnetic

Demagnetization is an effective method for achieving extremely low temperatures, often reaching fractions of a Kelvin, by reducing the magnetic field from its maximum value to zero while maintaining constant entropy.

3 Two thermally isolated ( d Qẳ0 andd Emẳ0) identical bodies A and B are at different temperatures TA[TB; their speciﬁc heat C is temperature independent The system of these two bodies is coming to thermodynamic equilibrium under the condition of extracting the maximum work through the help of a third body which undergoes cyclic operations What is theﬁnal temperature? What percentage of the energy lost by body A became work?

In a system approaching equilibrium, the total initial and final entropy must be equal due to the maximum work extracted By integrating the relation \( C = T \left( \frac{\partial S}{\partial T} \right) \) and considering that heat capacity (C) is temperature-independent, we can derive significant results regarding entropy changes.

S f ẳCẵln(T f =T 0 ị ỵln(T f =T 0 ị ỵ2S 0 where S 0 is the entropy of each body at the reference temperatureT0 By equating the two expressions for the total entropy we obtain

The energy lost by body A,CðTATfị, became partly workWand partly was transferred to body B, CðTf TBị: Thus W ẳCðTATfị CðTf TBị ẳ CðTAỵTB2Tfị ẳCð ffiffiffiffiffi

TB p ị 2 The percentage which became work is gẳ ð ffiffiffiffiffi

Unsolved Problems

1 The mass exchange termd Em in the general case of more than one kind of m-particles becomesd E m ẳP ilidN i In this case prove thatGẳP iliN i HintFrom the extensive/intensive discussion, we have the relation xGẳGðT;P;xN1;xN2; .ị

Take the derivative with respect toxand take into account thatdGị P ; T ẳd Em

2 The energyUof a system is decreasing as it is approaching its equilibrium state under the conditionsdS=dV=dN.Argue why this decrease does not violate the law of conservation of energy.

3 Prove (4.13) Hint: Take into account that DUẳ WỵQb eWb ! e )

WẳDUQ b e ỵW b!e ẳDUỵQ e b W e!b where Q e b ẳT0DS0

T0DS; W e!b ẳP0DV0ẳ P0DV Substituting these last relations in the expression for W and multiplying by −1 we have W DUþT0DS

P0DV ẳ DðUỵP0VT0Sị ẳAinitialAfinal

The Three Phases of Matter (Solid (s), Liquid (l), Gas (g))

The following page presents a graph illustrating the relationship between Gibbs free energy and temperature for the three phases of matter at constant pressure The analysis is categorized into three pressure conditions: (a) very low pressure, (b) intermediate pressure, and (c) very high pressure This exploration aims to justify the observed relationships and the subsequent plots.

GsðPị\GlðPị GgðPị; T ẳ0; SsðP;Tị\SlðP;Tị\SgðP;Tị ð4:26ị

In the analysis of phase transitions, the lower value of Gibbs free energy (G) at temperatures between 0T and Ts indicates the solid phase as the equilibrium state At the temperature Ts, a direct transition occurs from the solid phase to the gas phase, making the gas phase the lowest G phase for temperatures above Ts As pressure increases, the Gibbs free energy of the gas phase rises significantly faster than that of the solid and liquid phases, which remain relatively unchanged This leads to a unique point known as the triple point (Ptr), where all three phases coexist Beyond this point, for temperatures ranging from 0 to Tm, the solid phase is the equilibrium state, while at Tm, both solid and liquid phases coexist, and for temperatures between Tm and Tb, the liquid phase becomes the equilibrium state.

At elevated pressure, known as the critical pressure (Pcr), the curves representing the liquid phase (Gl) and gas phase (Ggas) no longer intersect, but instead merge smoothly This phenomenon indicates that beyond the critical point, there is no distinction between the liquid phase and the highly compressed gas phase.

Summarizing this description which is based exclusively on (4.26,4.27) we have the following phase diagram in the planeT,P:

What is the physical reason for theTmcurve (on which the solid and the liquid phase coexist) to be almost vertical?

The Clayperon-Clausius formula describes the slope of coexistence curves between two phases on a graph, which can be derived by analyzing two adjacent points on the curve that share the same Gibbs free energy (G) This relationship can be expressed mathematically as dP/dT = S2 - S1, where dP represents the change in pressure, dT represents the change in temperature, and S2 and S1 are the entropies of the two phases.

Given that ice is lighter than water, how we can freeze water of pressure 1 atm and temperature 0.1°C? By increasing or decreasing the applied pressure?

1 L.D Landau, E.M Lifshitz, Statistical Physics, Part 1 (Pergamon Press, Oxford 1980)

2 E Fermi, Thermodynamics (Dover Publications, NY, 1956)

3 M.W Zemansky, R.H Dittman, Heat and Thermodynamics, 7th edn (McGraw-Hill, New York, 1997)

Dimensional Analysis: A Short-Cut to Physics Relations

Dimensional analysis is a powerful method for quickly deriving physics formulas by identifying the correct combinations of relevant quantities and physical constants This technique ensures that the dimensions of the derived quantity match those of the selected combination The chapter will provide several examples to illustrate the application of dimensional analysis, which will be extensively utilized throughout the book to enhance physical understanding and promote derivation over memorization of formulas.

Outline of the Method

Dimensional analysis enables us to ascertain how a specific quantity X is influenced by relevant physical constants and independent variables The key challenge lies in accurately identifying all the relevant quantities that affect the value of X Once these quantities are determined, the next step is to derive a more general combination of them that shares the same dimensions—specifically, the same powers of length, time, and mass—as the quantity in question.

In both the Gauss-CGS and SI systems, the units of any physical quantity can be expressed in terms of fundamental quantities: length, time, mass, and, in the SI system, electric current This relationship can be mathematically represented as \(X = a \cdot \text{time unit}^m \cdot b \cdot \text{length unit}^n \cdot c \cdot \text{mass unit}^p\), highlighting the essential role of these fundamental units in defining physical quantities.

1 One main difference between the two systems is the way Coulomb ’ s law is written: in G-CGS

Fẳq1q2=r 2 ;while in SIFẳq1q2=4p e0r 2 wheree0is the permittivity of the vacuum. © Springer international Publishing Switzerland 2016

The dimensions of every physical quantity are defined by three numbers in G-CGS or four numbers in SI, where the additional number represents the power of electric current Specifically, in G-CGS, the dimensions are expressed as [X] = [time]^a [length]^b [mass]^c, and in SI, it includes current as well: [X] = [time]^a [length]^b [mass]^c [current]^d If a physical relationship exists in the form A = B or A[B or A\B, it follows that the dimensions of A and B must be equal, denoted as [A] = [B] Furthermore, when identifying all potential quantities A1 to An that the quantity of interest X may depend on, it is crucial that each quantity has distinct dimensions; if any quantities share dimensions, dimensionless ratios can be formed to maintain clarity.

In the G-CGS system, the variable ifn3 corresponds to orn4 in the SI system, leading to a unique combination represented by the form c1A l1 A ln, which shares the same dimensions as X This uniqueness allows for the determination of the formula for X, with the exception of the quantity c1.

Xẳc1A l 1 1 A l n n ð5:2ị where c 1 is a function of all dimensionless quantities (if there is none, c 1 is a numerical factor) and all n3 or n4 quantities A1; .An are of different dimensions.

Ifn[3 (in the G-CGS system) we choose three quantities A1;A2;A3 which deﬁne a system of units (in the sense that three different combinations of the form

A l 1 1 A l 2 2 A l 3 3 can be found which have the dimensions of length, time, and mass respectively) Then we have to perform the following steps:

(1) By a proper choice of m1;m2;m3 we form the combination A m 1 1 A m 2 2 A m 3 3 Xo, whereXo has the same dimensions as the quantity of interestX.

(2) We create also combinations A n 1 1n A n 2 2n A n 3 3n A no which have the same dimensions asAn, (nẳ4;5; .) and we deﬁne the dimensionless quantities

(3) Having determined the quantities Xo, A 40 , A 50 ,… in terms of the chosen quantities A1, A2, A3, we are ready to express X in terms of A1; .An as follows:

A50; . ð5:3ị where the unknown function f cannot be determined from dimensional analysis; additional information or even a complete physical theory is needed

68 5 Dimensional Analysis: A Short-Cut to Physics Relations toﬁnd out what f is A similar approach is followed in the SI system with n[3 replaced by n[4.

Several examples of the method of dimensional analysis are presented in this chapter demonstrating its extraordinary power.

Relations Regarding Some Eigenfrequencies

Determine by dimensional analysis the natural frequency (called eigenfrequency) of oscillation of a pendulum (Fig A.1a in Appendix A).

In the study of pendulum dynamics, key factors include the string length (l), mass (m), and gravitational acceleration (g), which influences potential energy; without gravity, the pendulum remains in equilibrium at any angle relative to the z-axis Additionally, the dimensionless maximum angle (h) between the string and the z-axis plays a crucial role The only combination of values for l1, l2, and l3 that yields dimensions equivalent to frequency is l1 = 1/2, l2 = 1/2, and l3 = 0 Consequently, the relationship can be expressed as x = √(g/l) f(h), where f(h) remains undetermined through dimensional analysis For small angles approaching zero, f(h) approaches a constant value, specifically f(0) = 1, as confirmed by detailed calculations.

Determine by dimensional analysis the natural frequency (called eigenfrequency) of oscillation of an LC circuit (Fig A.1b in Appendix A).

In an LC circuit, the key parameters are capacitance (C) and self-inductance (L), both measured in length units within the G-CGS system To derive the frequency dimension, we also need the speed of light (c), as electromagnetic phenomena are described by Maxwell's equations, which incorporate this velocity The relationship c = L gives a dimension of time to the power of minus one, aligning with frequency Notably, the ratio C/L is dimensionless, leading to the conclusion that dimensional analysis in the G-CGS system results in the expression x = f(c/L), where f represents an arbitrary function of the ratio C/L.

Let us try the other system, the SI, to see if it will do better than the G-CGS:

In the International System of Units (SI), the dimensions of capacitance and self-inductance are defined as the product of permittivity (ε) and length, and permeability (μ) and length, respectively To derive the dimensions of ε and μ, one can reference Coulomb’s law, which states that force (F) is proportional to the square of charge (q) divided by the square of distance (l), and the law of force per unit length between two currents From these principles, it can be concluded that the product of permittivity and permeability has dimensions of inverse velocity squared, while the ratio of permeability to permittivity corresponds to resistance squared Specifically, it can be expressed as ε₀l₀ = 1/c², highlighting the relationship between these fundamental physical constants.

The velocity of light in vacuum is denoted by c, and the impedance of the vacuum is represented by Z0, which is approximately 376.7 ohms In a circuit, the eigenfrequency can be determined by the values of inductance (L) and capacitance (C), and through dimensional analysis, it is found that the eigenfrequency is proportional to the square root of the product of L and C, expressed as x = √(1/LC).

; wherec 1 is a numerical factor which turns out to be one,c 1 = 1 Thus it follows that the function fðC=L) in the G-CGS result is equal to fðC=L)ẳ ffiffiffiffiffiffiffiffiffi

Some Relations in Fluid Dynamics

Poiseuille's formula describes the flow rate (Π) of an incompressible fluid through a circular pipe with radius r, factoring in friction This formula is essential for understanding fluid dynamics in pipes, as it quantifies the volume of fluid that passes through a given cross-section per unit time The flow rate is influenced by variables such as the fluid's viscosity, the pipe's length, and the pressure difference along the pipe By applying Poiseuille’s law, one can effectively analyze and predict the behavior of fluids in various engineering and scientific applications.

To maintain a steady flow in a pipe with friction, a pressure difference \( dP(l) \) between the pipe's ends is essential, which increases with the length \( l \) of the pipe This pressure difference is directly proportional to the length due to uniform friction forces acting along the pipe The relevant factor is the pressure gradient \( dP(l)/l \), which counteracts friction, a property influenced by the fluid's viscosity \( \eta \) Friction arises from the velocity gradient within the fluid, where the maximum velocity occurs at the center of the pipe and decreases to zero at the walls Consequently, the friction force per unit area is proportional to both viscosity and the velocity gradient Thus, the behavior of the system is determined by three parameters: \( dP(l)/l \), \( \eta \), and \( r \) Following the derived relationships, we establish that \( n = 1 \), \( m = 1 \), and \( k = 4 \).

Dimensional analysis serves as an efficient method for understanding relationships in physics, revealing that the numerical constant c1 is calculated to be π/8 Interestingly, the flow rate of liquid through a pipe is proportional to the square of its cross-sectional area, contrary to the intuitive expectation that it would be proportional to the area itself.

Derive the formula for the drag force acting on a solid object as it moves with a constant velocityυwithin afluid of viscosityη.

The drag force acting on a solid body is influenced by its velocity, size, shape, and the fluid's properties, including viscosity and density To maintain motion, energy must be expended to overcome friction, which arises from two distinct physical mechanisms.

The drag force experienced by a body moving through a fluid is influenced by the velocity gradient present in the fluid, particularly in scenarios like flow in a pipe This gradient arises because the fluid near the body moves with its velocity, while the fluid farther away remains at rest The frictional force resulting from this velocity gradient is directly proportional to the fluid's viscosity (η), the body's velocity (υ), and the cross-sectional area (A) of the body, which relates to its size and shape Therefore, the drag force can be expressed through a specific mathematical relationship that incorporates these factors.

In equation (5.6), the equality of dimensions on both sides results in the exponents a = 1, b = 1, and c = 1/2 The dimensionless numerical factor c1 varies based on the shape of the body For a sphere with radius r, the value of c1 is 6π when A1 is set to 2πr.

Friction arises from the kinetic energy continuously supplied to a fluid by a moving body, with the body's velocity sustained by external work countering drag forces This drag force is influenced by the fluid's density (ρ), the body's cross-sectional area (A), and its velocity (υ), as the kinetic energy per unit volume of the fluid is proportional to the density and the square of the velocity.

The equality of dimensions in equation (5.7) yields the exponent values k = 1, m = 2, and n = 1 The dimensionless numerical factor, commonly expressed as \( \frac{1}{2} C_D \), is influenced by the dimensional ratio \( \frac{g}{q \sqrt{pA}} \) and the shape of the object As this ratio approaches zero, for a sphere with radius \( r \) and area \( A = \pi r^2 \), the drag coefficient \( C_D \) approximates 0.48.

The ratio of the two types of drag forces (by omitting the coefﬁcientsc1; c2) is the so-called Reynolds numberRe:

F1ẳ q t 2 A g t ffiffiffi p ẳA t ffiffiffi pA g=q ẳt ffiffiffi pA m ð5:8ị

The Reynolds number is defined as the inverse of the ratio \( g = \frac{m}{q \sqrt{pA}} \), where \( m \) represents dynamic viscosity and \( q \) is density Kinematic viscosity, denoted as \( \nu \), is calculated as \( \frac{m}{q} \) For water, the dynamic viscosity is 0.001 kg m\(^{-1}\) s\(^{-1}\), resulting in a kinematic viscosity of 1 x 10\(^{-6}\) m\(^2\) s\(^{-1}\) In contrast, air has a dynamic viscosity of 1.81 x 10\(^{-5}\) kg m\(^{-1}\) s\(^{-1}\) and a kinematic viscosity of 1.51 x 10\(^{-5}\) m\(^2\) s\(^{-1}\) Additionally, the viscosity of blood at 37°C is measured at 2.7 x 10\(^{-3}\) kg m\(^{-1}\) s\(^{-1}\).

In the study of drag forces, Equation (5.8) indicates that for large bodies and high velocities (Re > 1000), the second mechanism prevails, while the first mechanism is more significant for small bodies and low velocities (Re < 0.2) In the intermediate range (0.2 < Re < 1000), where both mechanisms play a role, the total drag force is typically expressed as in Equation (5.7), with the coefficient c2 represented as 1/2 CD Here, CD is a function of the Reynolds number, beginning at a value proportional to 1/Re at the lower end of this range and decreasing monotonically as the Reynolds number increases.

Re1000 where the second mechanism fully dominates.

Thermodynamic Relations Revisited

Dimensional analysis, along with the extensive and intensive nature of thermodynamic quantities, enables us to refine their formulas For instance, the Gibbs free energy, which depends on two intensive variables (temperature and pressure) and one extensive variable (particle number), can be expressed in a specific functional form.

The function \( G \) is defined as proportional to \( N e \), where \( e \) represents a characteristic energy per particle, ensuring that \( G \) remains an extensive quantity with energy dimensions This relationship holds true as long as the other independent variables are intensive and dimensionless, preserving the extensive nature of \( G \) Additionally, \( a_3 \) signifies a characteristic volume per particle We anticipate that \( e \) and \( a \) will correlate with universal constants like \( h \) and \( e \), especially in scenarios where electromagnetic interactions are significant, as well as with particle characteristics such as mass \( m \).

Entropy (S) is an extensive thermodynamic quantity that depends on the number of particles (N), internal energy (U), and volume (V), and it shares the same dimensions as Boltzmann's constant (kB) This relationship highlights the extensive nature of entropy and its significance in thermodynamic analysis.

72 5 Dimensional Analysis: A Short-Cut to Physics Relations

In an ideal atomic or molecular gas, characterized by the absence of interactions, the relationship between the parameters e and a can be expressed as ea²/h² = m This indicates that the combination h² = m is significant, along with the natural independent variables and kB, as outlined in equations (5.9) and (5.10).

Waves in Extended, Discrete or Continuous, Media

Determine by dimensional analysis the velocity of propagationt(phase velocity) of a longitudinal wave in the system of coupled pendulums (see Fig A.2c in Appendix A).

The phase velocity \( t \) is influenced by various factors, including the lengths \( l \), mass \( m \), spring constant \( j \), acceleration due to gravity \( g \), and wavenumber \( k \) The potential energy has two contributions: one from gravity and the other from the springs, leading to an additive relationship in frequency squared: \( x^2 = x^2_g + x^2_j \) The frequency squared due to gravity is \( x^2_g = g/l \), while \( x^2_j \) depends on \( j \), \( m \), \( a \), and \( k \), with the general expression being \( x^2_j = (j/m)f(ak) \) Dimensional analysis shows that \( f \) is a positive function of \( ak \) that approaches zero as \( k \) approaches zero, indicating all masses move in phase The maximum value of \( f \) occurs when neighboring masses are out of phase, specifically when \( k = 2/a \) or \( ak = \pi \) The function \( f \) is expected to be proportional to \( \sin^2(ak/2) \), leading to the final expression \( x^2 = x^2_g + c_1 j m \sin^2(ak/2) \).

A thorough analysis reveals that the numerical constant \( c_1 \) is equal to 4 In a gravity-free environment and for \( k = 1 \) (where wavelengths significantly exceed \( a \)), the relationship \( x \approx \sqrt{\frac{j}{m} \cdot a \cdot k} \) simplifies to \( t = \frac{x}{k} \), indicating that the wave's propagation velocity is determined solely by the system's characteristics, independent of the wavelength.

The one-dimensional (1-D) analog of the bulk modulus, which represents the inverse of compressibility, is denoted as B, while q signifies the 1-D mass density It is noteworthy that the sound velocity in fluids is accurately expressed by the formula derived in this context.

Dimensional analysis can be used to determine the velocity of sea wave propagation by considering two contributions to potential energy: gravitational potential energy and surface tension potential energy The additivity of these energies leads to the relationship x² = x²g + x²s, where x²g is influenced by gravitational acceleration g, wavenumber k, and sea depth d, while x²s is dependent on surface tension coefficient r, wavenumber k, sea depth d, and density q The expressions for these terms can be formulated as x²g = gkfg(kd) and x²s = (rk³/q)fs(kd), with both fg(kd) and fs(kd) approaching a constant value of one when kd is large Ultimately, this yields the equation t²x²k² = gk + (rk/q), highlighting the relationship between wave velocity and the contributing factors.

At very long wavelengths, the influence of x²s becomes negligible compared to x²g, as x²s is derived from the relationship x²g / 1 = k² In this context, the sea depth will take precedence over other lengths, highlighting the significance of k in determining relevant measurements.

Dimensional analysis serves as an effective shortcut for establishing relationships in physics In this context, it is essential for fg(k, d) to be proportional to k^d in order to remove the k-dependence from the variable υ Remarkably, the proportionality constant is determined to be one, leading to the conclusion that t² is equal to g(d, k) multiplied by k raised to the power of d², as expressed in the equation t² = g(d, k) * k^d².

The velocity of tsunami propagation, as described by Equation (5.15), is influenced by the relationship between wavelength and sea depth, with the wavelength typically being much larger than the depth A detailed analysis reveals that the function fg(k) is equivalent to fs(k) = tanh(kd), confirming the two limits derived from basic physical principles Figure 5.1 illustrates the relationship between wave velocity and wavelength, noting that the surface tension coefficient for water at 293 K is 0.073 J/m² This graph clarifies that when the wind begins to blow, it initially generates sea waves with wavelengths around centimeters.

Summary of Important Formulae

The total drag force acting on a solid body of cross-sectional areaAmoving with velocityυwithin a fluid of densityρand viscosity ηis given by the formula

FDẳ 1 2 CDqAt 2 ð5:16ị whereC D is a dimensionless quantity which in general is a function of the so-called Reynolds number Ret ffiffiffi pA

=m; mg=q: For small values of the Reynolds numberðRe1ị, CDẳ2c1=Re, as to recover (5.6), while forRe1000,CDdoes

Sea wave velocity is influenced by wavelength, with very short wavelengths being dominated by surface tension In contrast, long wavelength waves, such as tsunamis, travel at high speeds; for instance, a wave with a depth of 2.5 km can propagate at 158 m/s, equivalent to 569 km/h The minimum velocity, approximately 0.23 m/s or 0.84 km/h, remains independent of the Reynolds number, as indicated in recovery equation (5.7), although exceptions may occur at specific Reynolds numbers.

10 5 to 10 6 , the so-called drag crisis).

The eigenfrequency of a wave propagating in the system of coupled pendulums (see Fig A.2c in Appendix A) depends on the wavevectork2p=kas follows: x 2 ẳx 2 g ỵ4 j msin 2 ð a k

In the absence of gravity, the equation for k=1 simplifies to x=k, leading to the relationship between velocity and parameters where velocity is expressed as \( t x=k = \sqrt{\frac{ja}{q}} \) This formulation is further generalized to derive the velocity of sound in fluids, represented as \( c_{sound} \).

ffiffiffi B q s ð5:18ị whereBis the bulk modulus defined asB Vð@P=@VịS.

Equation (5.17) plays a crucial role in understanding wavy ionic motions in solids in the absence of gravity Additionally, as discussed in Sect 12.8 of Chap 12, it can also be applied to describe wavy electronic motion through specific analogies This highlights how a single classical equation, specifically that of coupled pendulums, serves as a fundamental basis for exploring the quantum behavior of both electrons and ions in the solid state of matter.

Solved Problems

1 Obtain the thermodynamic quantities for a perfect monoatomic gas.

Entropy is fundamentally linked to the number of microscopic states within a system When analyzing a gas, we can consider a single atom among the N atoms present The classical state of this atom is defined by its position and momentum, leading to a number of possible states that is proportional to the product of the volume in real space and the volume in momentum space, represented as Vp³ The maximum momentum is associated with the maximum energy an atom can possess, which correlates with the average energy per particle Consequently, the number of possible classical states for a single atom is proportional to V(mU/N)^(3/2) Since each state of one atom can combine with any state of another, the total number of states for N atoms is C(N) = C(1)ⁿ Integrating this with the definition of entropy provides a comprehensive understanding of the system's behavior.

SẳNkBlnfconst:VðmU=Nị 3 = 2 g However, according to (5.10) the variable

To achieve the necessary dimensionless combination, V must be expressed as V = N a^3, while the variable ðmU = Nị^3 must equal 2 in the dimensionless form ðmU = N meị^3 = 2 The additional factors N 1 ðmea^2 ị^3 = 2 possess dimensions of (energy time) −3, making them proportional to h^3, the only relevant universal constant with dimensions of energy time By applying these essential corrections to the classical result, we derive the final outcomes.

N þ 3 2 ln c1 m h 2 ð5:19ị wherec 1 is a constant numerical factor which turns out to be equal to e 5 = 3 =3p; and e is Euler’s number, e = 2.7182….By taking the derivative ofSwith respect toUwe obtain

By taking the derivative ofS with respect toVwe obtain the pressure

The Helhmoltz free energyFcan then be determined fromF=U−TSand the Gibbs free energy from G =F+PV.(The reader may easily obtain the speciﬁc heats as well).

The dimensional correction factor \( N_{1} \) is influenced by Quantum Mechanics, as indicated by the presence of Planck's constant \( h \) Unlike classical physics, which defines a particle's state by a single point in both real and momentum space, quantum mechanics requires consideration of the uncertainty principle This principle dictates that a single state is characterized by a non-zero volume in real space \( \Delta r^3 \) and momentum space \( \Delta p^3 \), with their product approximately equal to \( h^3 \) Consequently, the number of states \( C(1) \) for a single atom, in light of Heisenberg's uncertainty principle, is expressed as \( C(1) = n V m U N^{3/2} \).

Due to Pauli's principle, identical atoms sharing the same volume are indistinguishable, leading to the conclusion that any permutation of the N atoms does not alter the count of microstates, C(N) Consequently, the classical approach overestimates the number of microstates by a factor of N! Therefore, the actual number of microstates, in accordance with Pauli's principle, is given by C(N) = C(1) * N / N!, where C(1) represents the count for a single atom.

78 5 Dimensional Analysis: A Short-Cut to Physics Relations account thatN! ðN=eị N where e is Euler’s number, e = 2.7182… Combining these quantum corrections we obtain again (5.19).

2 Estimate the life-time of a classical model of a hydrogen atom Assume that, if there were no radiation, the electron will follow a circular orbit of radius aB.

In a hydrogen atom, an electron acting as a classical particle will ultimately spiral into the proton due to energy loss from electromagnetic radiation The time it takes for the electron to reach the proton is influenced by several factors: the radius of its initial circular orbit (aB), the mass of the electron (me), the charge of the electron (e), and the speed of light (c) By combining these three quantities—radius, mass, and charge—we can derive a time dimension in the Gaussian-CGS system, represented as a function of aB, me, and e.

. e and another one with dimensions of velocity, toẳaB=toẳe

. a 1 B = 2 m 1 e = 2 Hence, the most general expression for the timet ‘ has the form,t ‘ ẳtof t c o ;wherefðto=cịis an arbitrary function of the dimensionless variableto=cẳe=a 1 B = 2 m 1 e = 2 c:

The lifetime \( t' \) is inversely proportional to the radiated power, which is in turn inversely proportional to the cube of the speed of light This relationship can be expressed as \( f(to/c) = b \cdot c^3 \cdot t^3_o \), where \( b \) is a numerical factor Consequently, the formula for \( t' \) is given by \( t' = ba^3 B m^2 e c^3 e^4 \).

The periods of classical electronic motion without electromagnetic radiation can be derived from Newton's equation of motion This equation can be expressed as \( m e x^2 r^2 = e^2 r \) or, at \( t = 0 \), as \( x^2 = \frac{e^2}{m e a^3 B} \) Consequently, the relationship can be simplified to \( s^2 p x = 2p m e a^3 B e^2 \).

To derive an explicit result for the time variable \( t' \), we begin with the equation for energy loss due to radiated electromagnetic power \( I \) from the electron, represented as \( \frac{d}{dt} \left( \frac{e^2}{2r} \right) = I \) According to the subsequent chapter, \( I \) can be expressed as \( \frac{2}{3} \frac{e^2 a^2}{c^3} \) This equation holds under the assumption that the electron's fall towards the center is a relatively slow process compared to the period \( s \), leading to the relationship \( e_{\text{total}} = e_P = 2 \) for each revolution With this condition, the acceleration \( a \) is primarily centripetal, given by \( a = \frac{e^2}{m r^2} \) By combining these equations, we derive the following equation for \( r \): \( \frac{dr}{dt} = \frac{4}{3} \frac{e^4}{m^2 e c^3} r^2 \), which simplifies to the solution \( r^3 = a^3 \left( \frac{4 e^4}{m^2 e c^3} t \right) \).

By settingrẳ0 weﬁnd that t ‘ is given by (5.19) with the numerical factorb being equal to 1=4 We must check whethert ‘ =s1 Working in atomic units (eẳm e ẳa B ẳ1;c137) we ﬁnd that indeed t ‘ =sẳ c 3

3 Estimate the lifting force on the wings of a plane Assume that the wings are rectangularflat rigid metallic sheets.

The lift force (F') on an aircraft's wings is influenced by the wing area (S), the plane's velocity (t), and the presence of air, as lift cannot occur without it Key air properties such as density (q) and viscosity (g) also play a role, although viscosity's impact diminishes at high speeds Additionally, the lift force is affected by the angle of attack (α), which is the angle between the wings and the plane's velocity vector At zero angle of attack, the lift force is zero, but for small angles, the lift can be approximated as proportional to the angle of attack.

F ‘ ẳc1qSt 2 / ð5:26ị wherec1is a numerical factor In reality the shape of the wings, which can be adjusted by the position of theflaps, determine an effective angle/.

Unsolved Problems

1 Obtain the speed of sound in air.

In the context of air, the term \( \frac{3}{2} \ln\left(\frac{U}{N}\right) \) in equation (5.19) should be modified to \( \frac{5}{2} \ln\left(\frac{U}{N}\right) \) due to the presence of diatomic molecules such as \( N_2 \) and \( O_2 \) These diatomic molecules possess three translational and two rotational degrees of freedom By substituting \( \frac{U}{N} \) with \( \frac{3}{2} \frac{PV}{N} \) in equation (5.19), one can then compute the differential of entropy \( S \) to derive the necessary results.

2 For a car moving with 108 km/h estimate how much power is needed to overcome the air resistance (1 hp = 746 W).

3 Raindrops have a diameter usually between 0.5 to 2.5 mm (it can even reach

5 mm in thunderstorms) Estimate their terminal speed.

Interactions

Structures Held Together by Strong Interactions

Structures Held Together by Electromagnetic Interactions

Gravity at the Front Stage

Tiêu đề	From Quarks to the Universe A Short Physics Course
Tác giả	Eleftherios N. Economou
Trường học	University of Crete
Chuyên ngành	Physics
Thể loại	book
Năm xuất bản	2016
Thành phố	Iraklion

Định dạng
Số trang	323
Dung lượng	17,98 MB