What Is a Classical Entity?
In classical physics, the concept of an object is based on the belief in an objective reality, where a material entity possesses intrinsic properties that are inherent to itself The relationships between classical objects stem from this objective reality, ensuring that the properties of these entities remain constant and do not rely on external factors, including the act of observation or measurement.
With the development of Maxwell’s equation and Einstein’s theory of gravity, the classical concept of a physical entity was extended to include the classical field.
A classical field, such as the electromagnetic field, is a physical entity that is spread a b
The classical perspective posits that the existence of an apple is an objective reality that remains independent of the observer's perception This reality exists consistently across space and time, with the classical field behaving like a tangible entity It possesses intrinsic properties, including energy and momentum, at every point in spacetime it occupies.
Classical physics posits that entities are entirely determinate and exist in precisely defined states, such as a classical particle having a specific position in space Upon observation, these classical entities manifest as they truly are, making them wholly empirical; thus, an observation can fully and accurately describe their state The Oxford dictionary defines "empirical" as being based on or verifiable through observation or experience, rather than relying solely on theory or pure logic.
We conclude that a classical entity exists objectively and is a determinate quantity.
In classical physics, the existence of objects is objective, meaning that the state of the apples remains unchanged regardless of whether a person is looking at them or not Thus, even when the individual looks away, the apples continue to exist in the same state in both scenarios.
If an observer is not looking at the apples, there is no experimental evidence to assert that the apples remain in the same state as when they were last observed This highlights that classical physics' assertion of an objective reality is fundamentally an assumption.
The behavior of classical dynamical variables, such as position and velocity of a particle, is governed by Newton's second law of motion In contemporary physics, this principle is expressed through the variation of the system's action, S, which encapsulates the time evolution of these variables.
In classical mechanics, the trajectory of a particle is defined by its position x(t) and velocity v(t) = dx(t)/dt To fully describe the classical state of a particle at any given moment, it is essential to specify its dynamical variables, which include both position and momentum p.
Reserved) action is the time integral of the LagrangianL(x,dx/dt), which is a function of the kinetic and potential energy of a particle’s trajectory.
In the finite time interval [t_i, t_f], the particle's trajectory, illustrated in Fig 2.2a, is governed by Newton's law of motion At any given moment t, the particle possesses a specific position x and momentum p, defined as p = mv, where v = dx/dt represents its velocity and m denotes its mass Once the trajectory of the particle is established, the Lagrangian L(x, dx/dt) can be calculated.
The action S for a particle is given by the following:
⇒δS=0 : Equation of motion (2.2) with the initial and final positions being specified at t iand t f, respectively.
The equation of motion expressed in (2.2) indicates that for any chosen trajectory, the calculated value of S achieves a minimum or maximum only when the trajectory complies with Newton's law, resulting in δS=0 Therefore, we can infer that (2.2) is fundamentally equivalent to Newton's law.
In classical mechanics, the state of a particle is defined by its dynamical variables, specifically position (x) and momentum (p), which remain constant at any given moment in time (t) This means that the particle's classical state is fully determined by these variables, providing a precise description of its condition, as illustrated in Fig 2.2b.
An apple is made up of numerous atoms, and it's unlikely that all of these atoms remain with the apple over time While not observing it, some atoms may have detached from the apple, while others from the surrounding environment may have attached themselves, indicating a constant exchange of atomic components.
Describing an apple, like any large classical system, involves inherent approximations, similar to the challenge of accurately determining the exact position and momentum of all gas atoms in a room Precision, as defined previously, refers to the consistency of repeated experiments yielding measured values within a specific error margin An approximation, therefore, entails expressing a quantity with a clearly defined degree of precision.
The emergence of classical chaos theory highlights that all practical measurements possess finite accuracy, leading to the concept of classical randomness in the analysis of nonlinear classical systems.
Classical randomness refers to phenomena characterized by unpredictability and chance, which can be mathematically represented through variables defined by their probability distribution functions.
Nevertheless, classical chaos theory does not change the ontological property of a classical system in that it exists in a determinate and intrinsically exact state.
Ontology: from the Greek term for ‘being’; that which ‘is,’ present participle of the verb
‘be’; the term is used for the nature of being, of existence, or of reality.
The uncertainty and lack of clarity in understanding a large classical system or chaotic process stem from our limited knowledge of its precise state, which ultimately gives rise to classical probability theory.
The Entity in Quantum Mechanics
Our discussion on the “entity” in quantum mechanics does not take the historical route but rather starts from the quantum conception of the entity and then goes on to a b
The inherent indeterminacy of an atom means that its properties cannot be precisely defined until observed, as illustrated in Fig 2.3 In part (a), the shaded volume represents the area where the atom is most likely to be detected In part (b), the act of observation forces the atom into a determinate state This phenomenon highlights the necessity for a mathematical framework in quantum mechanics, which effectively captures the behavior of quantum entities and their transition from indeterminate to determinate states upon measurement.
In a quantum particle confined within a box, repeated measurements of its position reveal distinct outcomes each time Initially, measuring the particle's position yields a specific location, x1 However, subsequent identical measurements result in different positions, such as x2 and x3, demonstrating the inherent unpredictability of quantum mechanics Each measurement, despite the identical preparation of the particle, consistently produces a unique position, highlighting the fundamental nature of quantum behavior.
In Section 5.3, we explore unique quantum states known as eigenstates, which possess consistent properties like energy and angular momentum that remain unchanged upon observation However, the position degree of freedom for a particle in a box does not exhibit this same characteristic.
In the realm of quantum mechanics, unobserved particles, such as electrons or atoms, lack a definite position and do not possess objective existence, unlike classical particles While larger entities, like stones, can be adequately described by classical physics, the behavior of small particles challenges our intuitive understanding, as their observed and unobserved states diverge significantly.
When observing an atom directly, it appears as a point-like object; however, in the absence of observation, quantum mechanics reveals that the atom lacks a definite position Instead, it exists in multiple locations simultaneously, each with varying probabilities The area of highest likelihood is illustrated in the shaded region of the diagram, where the degree of shading reflects the atom's positional probabilities.
Section 9.3 explores the concept of repeated quantum measurements, highlighting the observation of particles at various locations throughout space The terms "indeterminate" and "quantum uncertainty" are often used interchangeably; however, a more precise definition of "indeterminate" is provided in Section 3.2.
Quantum particles differ from classical particles in that they lack a definite position until observed This fundamental distinction between the observed and unobserved states of quantum entities lies at the core of quantum mechanics Numerous experiments have consistently validated this intriguing aspect of quantum theory.
The central role of observation, of measurement, is what differentiates the observed from the unobserved state and is the key to quantum mechanics.
Heisenberg introduced the concepts of potentiality and actuality in quantum mechanics, where potentiality refers to the indeterminate state of a quantum entity, and actuality denotes its observed condition Each observation causes a quantum particle to shift from its potential state to a specific, actual determinate state.
In quantum mechanics, a quantum entity exists in two distinct states: it is definite and determinate when observed, and indeterminate and uncertain when unobserved The crucial link between these two states is the act of observation, which serves as the measurement process that transitions the entity from its unobserved to its observed form.
Describing an Indeterminate Quantum Entity
A classical entity is defined by its dynamical variables, which outline its state, and is further characterized by the equations of motion that govern these variables.
The classical approach falls short in accurately describing quantum entities due to their intrinsic indeterminacy To effectively characterize a quantum particle, it is essential to address several interrelated issues arising from quantum mechanical indeterminacy.
In quantum mechanics, describing a quantum particle begins with the quantum generalization of classical dynamical variables, a fundamental shift from classical physics Unlike classical particles, quantum particles do not follow a defined trajectory due to inherent indeterminateness, rendering it impossible to precisely know both position and momentum simultaneously This concept, a cornerstone of quantum theory, necessitates a trade-off between measuring a particle's position and its momentum, as determining one aspect precludes knowledge of the other.
• Since the quantum entity’s position is indeterminate, the classical particle’s dynamical variables x and p are superseded by the quantum degree of freedom
F For a quantum particle moving in one space dimension, the degree of freedom space is given by the real line, namely,F=ℜ={x|x∈[−∞,+∞]}, and hence,
In quantum mechanics, a particle's position remains uncertain when unobserved, necessitating the use of operators to measure its degree of freedom These operators, particularly position projection operators, are essential for capturing the effects of the particle's position, as elaborated in Chapter 5 and Section 9.2.
Repeated measurements of a quantum particle's position operators provide insight into the potential values for the particle's position This process enables the theoretical enumeration of all possible locations of the particle Consequently, the outcomes of these observations facilitate the mathematical reconstruction of the particle's degree of freedom space, denoted as spaceF.
• The quantum degree of freedom is a quantitative entity that numerically describes all the possible allowed values for the quantum entity and constitutes the space
F The degree of freedom is a time-independent quantity, with the spaceFbeing invariant and unchanging over time.
Experimental evidence shows that when quantum particles are repeatedly observed using various position projection operators, these operators yield different average values that reflect the particle's degrees of freedom Such repeated observations not only enable the enumeration of the degree of freedom \( F \) but also indicate the probability of locating the particle across different position projection operators.
A significant conceptual advancement, inspired by Max Born's work, suggests that the outcomes of repeated experimental observations of the state vector can provide a comprehensive understanding of all the quantitative properties of a quantum entity.
The quantum state vector ψ(F) offers a quantitative and probabilistic description of indeterminate quantum entities This state vector is inherently statistical, making every outcome entirely unpredictable It serves as an element within the state space of the degree of freedom, represented as V(F).
• The quantum state vectorψ(F)is postulated to carry a complete description of the quantum entity and is a superstructure of the quantum degree of freedom
The state vector ψ(F) plays a crucial role in determining the probability of specific experimental outcomes related to the observed degrees of freedom Additionally, the quantum state can be expressed through the density matrix operator, which is particularly effective for examining quantum measurement processes, as elaborated in Chapter 6.
The dynamics of a quantum entity is governed by the time evolution of its state vector, ψ(t,F), which explicitly shows its dependence on time (t) The Schrödinger equation, a first-order partial differential equation in time, describes this evolution and provides the rate of change of the state vector, represented as ∂ψ(t,F)/∂t.
In quantum mechanics, entities exist in two states: potential (unobserved) and actual (observed), which are linked through the measurement process This relationship highlights the concept of indeterminateness as potential and determinateness as actual.
2 Quantum probability is different from classical probability and is discussed in Chap 7.
3 The relation of the state to observed quantities is discussed in Sect 2.4.
The theoretical framework of quantum mechanics describes a quantum entity characterized by its degree of freedom F and a state vector within the state space V(F) Operators O(F) interact with the state vector to derive information regarding the degree of freedom, ultimately producing a final result E[V(O(F))] Notably, only this final result, which is distanced from the quantum entity itself, is subject to empirical observation.
Werner Heisenberg posits that all physical properties associated with the degree of freedom F can be expressed through mathematical operators O(F) In quantum mechanics, the measurement process is depicted by the application of these operators to the quantum state ψ(F) of the quantum entity.
• Repeating the process of measurement results in the experimental determination of the average, or expectation value, of the physical operators and is expressed as
E ψ [O(F)] All physical information about the degree of freedomF is encoded in the expectation value of operators.
In conclusion, a quantum entity is significantly more complex than a classical entity, as illustrated in Fig 2.4 When unobserved, a quantum entity comprises its degrees of freedom (F) and the state vector (ψ(t,F)), which together define its observable properties.
In classical physics, an entity's observable condition fully reveals its properties, meaning "one sees what one gets." However, for quantum entities, there exists an unobservable superstructure that separates the empirically observed properties from the complete essence of the quantum entity This hidden quantum superstructure, which does not exist in classical entities, requires interpretation to link the quantum entity with its observed properties, a connection established by the Copenhagen quantum postulate.
The Copenhagen Quantum Postulate
The Copenhagen interpretation of quantum mechanics, developed by Niels Bohr and Werner Heisenberg, is the predominant view adopted by most physicists and serves as the foundation for this book Sections 2.2 and 2.3 provide a comprehensive overview and clarification of the core principles of this interpretation.
The Copenhagen interpretation of quantum mechanics is not universally accepted within the physics community, as numerous alternative explanations have been proposed This book aims to clarify the theoretical assumptions underlying the Copenhagen interpretation, which are often obscured by the complex mathematical formalism of quantum mechanics.
The Copenhagen interpretation can be summarized by the following postulate:
A quantum entity is defined by its degree of freedom, F, and its state vector, ψ(t, F) The degree of freedom represents a range of values that form a space, F, while the quantum state, ψ(t, F), is a complex-valued function that fully describes this degree of freedom This state vector is an essential element of the state space, V(F), characterizing the quantum entity's behavior and properties.
All physically observable quantities are obtained by applying Hermitian operators O ( F ) on the state ψ( t , F ).
The quantum entity is an inseparable pair, namely, the degree of freedom and its state vector.
Experimental observations collapse the quantum state and repeated observations yield
The expectation value of the operator O(F) for the state ψ(t, F) is represented as Eψ[O(F)] While the Schrödinger equation governs the time evolution of the state vector ψ(t, F), it does not dictate the measurement process.
It needs to be emphasized that the state vectorψ(t,F)provides only statistical information about the quantum entity; the result of any particular experiment is impossible to predict.
The organization of the theoretical superstructure of quantum mechanics is shown in Fig.2.4.
The quantum state ψ(t,F) represents a complex number that captures the degree of freedom, serving as a fundamental element beyond the real positive numbers observed as probabilities This framework for assigning expectation values to operators, exemplified by E ψ [O(F)], facilitates the transition from classical to quantum probability, a topic explored in depth in Chapter 7.
The Copenhagen quantum postulate can be illustrated through a quantum particle moving in one dimension along the real line, represented as F=ℜ={x|x∈(−∞,+∞)} with the state function ψ(t,ℜ) When a measurement is made using the position operator O(x), it projects the quantum state to a specific point x∈ℜ, resulting in the collapse of the quantum state.
Note from (2.3) that P(t,x) obeys all the requirements to be interpreted as a probability distribution A complete description of a quantum system requires
The position projection operator O(x) = |x⟩⟨x| is examined in Section 9.2, which defines the probability P(t, x) for all potential states of a quantum system For a quantum particle in space, its possible states correspond to various positions x within the range of [−∞, +∞] Consequently, the exact position of the quantum particle remains indeterminate.
P(t,x) =|ψ(t,x)| 2 is the probability of the state vector collapsing at time t and at
O(x)—the projection operator for position x.
The moment that the stateψ(t,ℜ) is observed at specific projection operator
O(x), the stateψ(t,ℜ)instantaneously becomes zero everywhere else The transi- tion fromψ(t,ℜ)to|ψ(t,x)| 2 is an expression of the collapse of the quantum state.
The collapse of a quantum state is a unique phenomenon that doesn't occur in classical waves, emphasizing its purely quantum nature In the early days of quantum mechanics, pioneers referred to it as "wave mechanics" due to the state ψ(t,ℜ) resembling a classical wave spread across all spaceℜ, replacing the traditional Newtonian description of particles by their trajectories x(t).
“wave function” was used for denotingψ(t,ℜ).
The state ψ(t,F) of a quantum particle differs fundamentally from a classical wave, sharing only the characteristic of occasional spatial spread Unlike classical waves, certain quantum states, such as the up and down spin states, are localized at a single point in space, exhibiting no spatial dependence and therefore not spreading over space.
In the text, the terms state, quantum state, state function, or state vector are henceforth used for ψ(t,F), as these are more precise terms than the term wave function.
The findings presented in (2.3) highlight a significant breakthrough in quantum theory, revealing that beneath observable experimental outcomes, which allow for the calculation of probabilities P(t,x) = |ψ(t,x)|², exists an unseen realm characterized by the probability amplitude, comprehensively defined by the quantum state ψ(t,F).
Five Pillars of Quantum Mechanics
The description and dynamics of a quantum entity given in Sect 2.3 can be summarized as follows Quantum mechanics is built on five main conceptual pillars that are given below.
• The quantum degree of freedom spaceF
• Time evolution ofψ(F)given by the Schrửdinger equation
• The process of measurement, with repeated observations yielding the expectation value of the operators, namely, E V [O(F)]
The five pillars of quantum mechanics are illustrated in Fig.2.5 Each pillar of quantum mechanics is briefly summarized in the following sections.
Fig 2.5 The five cardinal pillars of quantum mechanics (published with permission of © Belal
Degree of Freedom Space F
In classical mechanics, a system is characterized by its dynamical variables, with its time evolution governed by Newton's equations of motion In contrast, quantum mechanics introduces a new framework where the description of a quantum entity is based on a generalization of these classical variables, referred to as the quantum degree of freedom, as outlined by Dirac.
The degree of freedom serves as the fundamental basis for a quantum entity, encapsulating its essential qualities and properties For instance, an electron possesses multiple degrees of freedom, including spin, position, and angular momentum, which collectively define its characteristics The symbol F is used to represent all these degrees of freedom within a quantum entity.
Chapter 7 explores the indeterminacy of quantum degrees of freedom, revealing that these degrees do not possess precise values until observed, confirming their inherent indeterminacy This intrinsic uncertainty suggests that each degree of freedom exists within a range of potential values, collectively forming a space referred to as F, which remains constant over time.
The entire edifice of quantum mechanics is built on the degree of freedom and, in particular, on the spaceF.
State Space V (F )
In the quantum mechanical framework, a quantum degree of freedom is inherently indeterminate and, metaphorically speaking, simultaneously has a range of possible values that constitutes the spaceF.
An experimental device is designed to investigate the properties of a degree of freedom, specifically focusing on a quantum entity with spin that features 2 + 1 discrete points To effectively observe a spin system, the device must accommodate 2 + 1 distinct positions, corresponding to each possible value of the degree of freedom.
The quantum degree of freedom necessitates repeated experiments due to its inherent indeterminacy, as outlined in Sect 9.3 Each individual experiment yields uncertain and indeterminate outcomes, allowing the device to adopt any of its numerous potential values However, analyzing the cumulative results of these repeated experiments reveals a discernible pattern, with certain positions of the device pointer being more frequently observed than others.
Quantum probability emerged to address the statistical regularities of indeterminate outcomes in experiments involving degrees of freedom To capture quantum indeterminacy, a complex-valued state vector, known as the state function and represented by ψ, is utilized to describe observable properties This quantum state ψ maps a space F to complex numbers C, specifically indicating that for coordinates x within the real numbers ℜ, ψ functions as a complex function of x, expressed as ψ: F → C, where x ∈ ℜ implies ψ(x) ∈ C.
The state vector is a crucial component of an infinite-dimensional linear vector state space To ensure a coherent probabilistic interpretation of quantum mechanics, as outlined in Chapter 4, it is essential that the norm of the state vector ψ is equal to one.
The state vector hence is an element of a time-independent normed linear vector space, namely, Hilbert spaceV, which is the subject matter of Chap 4 In symbols ψ∈ V(F)
Operators O(F)
The relationship between quantum degrees of freedom and their observable properties is inherently indirect and relies on the measurement process A coherent interpretation of quantum mechanics necessitates that measurement is central to its theoretical framework.
In classical mechanics, the definition of a classical system is independent of observation and measurement of physical properties A classical particle is completely characterized by its position and velocity at a specific time, denoted as t.
In quantum mechanics, it is generally understood that a quantum state is not an eigenstate unless specified The position and velocity of a classical particle, represented as x(t) and v(t), exist objectively, independent of any measurements taken to determine them This highlights that the existence of these properties is not contingent upon observation.
In contrast to classical mechanics, in quantum mechanics, the degree of freedom
In quantum mechanics, the degree of freedom F cannot be directly observed; instead, observable physical properties arise from measurements performed on the state vector ψ Operators, which are mathematical constructs representing these physical properties, interact with the state vector to mathematically depict the measurement process According to Dirac, these operators that signify physical quantities are referred to as observables.
The degree of freedom (F) and its measurable properties, denoted by operators (O i), are linked through the quantum state vector (ψ(t,F)) Experiments can only assess the impact of this degree of freedom on the operators via the state vector Additionally, each experimental device is specifically engineered to measure a particular physical property of the degree of freedom, represented by the operator O i.
The Schrửdinger Equation for State ψ (t ,F )
The focus of this discussion has been on the kinematics of quantum systems, emphasizing the framework for their description A key objective in physics is to derive dynamical equations that forecast a system's future state In quantum mechanics, this is achieved through the Schrödinger equation, which governs the time evolution of the state function ψ(t,F), with 't' representing time Notably, the Schrödinger equation is time-reversible, underscoring its fundamental role in predicting quantum behavior.
To formulate the Schrödinger equation, it is essential to define the quantum version of energy The Hamiltonian operator, denoted as H, signifies the energy of a quantum system and dictates the structure and numerical spectrum of the permitted energy levels for that system.
All physical entities require energy for existence, which justifies the inclusion of the Hamiltonian H in the Schrödinger equation Energy, being conjugate to time—much like position is to momentum—suggests that H is crucial for the time evolution of the state vector However, ultimately, the Schrödinger equation lacks a derivation from fundamental principles and must be accepted as a postulate.
The celebrated Schrửdinger equation is given by
∂t =Hψ(t,F) (2.4) where ¯h=h/2π, with h being Planck’s constant Consider a quantum particle in one dimension; the degree of freedom is given byF=ℜ; for the coordinate x∈ℜ, the Schrửdinger equation is given by
The Hamiltonian, for potential V(x), is given by
The Schrödinger equation is a linear equation for the state function ψ, meaning that if ψ1 and ψ2 are solutions, then their linear combination ψ = αψ1 + βψ2 is also a valid solution, where α and β are complex numbers This principle of quantum superposition of state vectors has significant implications, which are explored in Section 3.7.
Quantum mechanics complicates the description of Nature by replacing classical mechanics' six real dynamical variables (x and p) with an entire space of indeterminate degrees of freedom This necessitates the use of a state vector that functions within this space However, Dirac suggests that the complexity introduced by quantum indeterminacy is balanced by the simplification provided by the linearity of the Schrödinger equation.
The state vector ψ belongs to a normed linear vector space, a consequence of the linearity inherent in the Schrödinger equation This characteristic of ψ facilitates various nonclassical phenomena, including the emergence of entangled states and the principle of quantum superposition, which will be explored in detail in Chapters 6 and 8.
Indeterminate Quantum Paths
Understanding the time evolution of physical entities is crucial for comprehending Nature In classical mechanics, the trajectory of an evolving entity is objective and independent of observation, as illustrated in Fig 2.2a At any given moment, both the position x(t) and velocity v(t) of the entity hold precise values, highlighting the deterministic nature of classical motion.
We need to determine the mode of existence of quantum indeterminacy for the case of the time evolution of a quantum particle.
In quantum mechanics, consider a particle represented by the degree of freedom x, which belongs to the real numbers F When the particle is first observed at time ti, the position operator identifies its location at point xi A subsequent observation occurs at time tf, where the position operator again determines the particle's position.
A quantum particle is first observed at an initial position \( x_i \) at time \( t_i \) and later at a final position \( x_f \) at time \( t_f \) The indeterminate nature of the quantum particle's path indicates that it exists simultaneously in all possible paths.
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In the study of quantum particles transitioning from an initial position \(x_i\) at time \(t_i\) to a final position \(x_f\) at time \(t_f\), we consider a system with \(N\) slits located at positions \(x_1, x_2, \ldots, x_N\) There are two scenarios: one where the intermediate path at time \(t\) is observed and another where it is not The implications of these scenarios are explored in detail in Chapters 8 and 11, with a specific focus on the two-slit experiment discussed in Section 3.7 When the paths are observed, the results align with classical predictions, as elaborated further in Chapter 8.
What is the description of the quantum particle making a transition from x i ,t ito x f ,t fwhen it is not observed at intermediate time t? The following is a summary of the conclusions:
• The quantum indeterminacy of the degree of freedom leads to the conclusion that the path of the quantum particle is indeterminate.
Quantum particles exhibit path indeterminacy by existing in all potential paths at once, effectively "taking" every possible route simultaneously.
• For the case of N-slits between the initial and final positions shown in Fig.2.6, the quantum particle simultaneously exists in all the N-paths.
The term probability amplitude is used for describing the indeterminate paths of a quantum system The probability amplitude is a complex number, and each determinate path is assigned a probability amplitude.
In the absence of observations to identify the specific path taken, all possible paths for the particle become indistinguishable, leading to an indeterminate state where the particle exists simultaneously across all N-paths The probability amplitude for this quantum particle, characterized by its indeterminate path, is derived by combining the probability amplitudes of the distinct determinate paths through the quantum superposition principle, which is elaborated upon in Section 3.7 for the two-slit scenario and further explored in Chapter 8.
In quantum mechanics, the probability amplitude φ n is associated with a specific path through a slit at position x n, where n ranges from 1 to N When considering a particle observed at position x i at time t i and later at position x f at time t f, the overall probability amplitude φ for this transition is derived by summing the probability amplitudes of all indistinguishable paths This relationship can be expressed mathematically as φ(x f ,t f |x i ,t i) =∑ N n=1φ n, highlighting the significance of indistinguishable paths in determining the particle's behavior.
Once the probability amplitude is determined, its modulus squared, namely,|φ| 2 , yields the probability for the process in question For the N-slit case,
|φ(x f ,t f |x i ,t i)| 2 =P(x f ,t f |x i ,t i) ; dx f P(x f ,t f |x i ,t i) =1 where P(x f ,t f |x i ,t i)is the conditional probability that a particle, observed at position x iat time t i, will be observed at position x fat later time t f.
Quantum mechanics can be expressed through indeterminate paths, a formulation that operates independently of the state vector and the Schrödinger equation This method, referred to as the Dirac–Feynman formulation, is explored in Chapter 11.
The Process of Measurement
Numerous experiments demonstrate that the readings obtained from observing a quantum entity using a measuring device cannot be explained by deterministic classical physics, highlighting the necessity of quantum mechanics for an accurate interpretation.
Consider a degree of freedomF; the existence of a range of possible values of the degree of freedom is encoded in its state vectorψ(F) Let physical operators
In quantum mechanics, O(F) represents the observables associated with the quantum degree of freedom, which cannot be directly observed Instead, measurements reflect the influence of this degree of freedom on operators, mediated by the state vector ψ(F) Preparing a quantum state results in the state ψ(F), which undergoes multiple measurements to analyze its properties.
A concrete example of how the quantum state of a “quantum particle in a box” is prepared is discussed in Sect 9.3, which we briefly review.
Electrons are generated by heating a metal, resulting in their ejection, which allows them to be treated as semiclassical particles These electrons are directed into a cavity using electric and magnetic fields To prevent the electrons from colliding with the cavity walls, a specialized structure known as a Penning trap is employed, which utilizes electric and magnetic fields to confine the electron's movement within a specific spatial region.
Once the electron is confined within the cavity, it remains undisturbed, with its energy set to keep it within the "box." The electron's behavior is characterized by the quantum state ψ(F), representing a quantum particle in a box Measurement theory relies on operators O(F), as discussed in Chapter 5, which serve as the mathematical foundation for measuring quantum properties To effectively measure these properties, one must understand specific quantum states known as the eigenstates χn of the operator O(F).
Measurement determines the properties of a system's degree of freedom by utilizing an experimental device Mathematically, this process is expressed by applying the operator O(F) to the system's state ψ(F), resulting in a projection onto one of the eigenstates of O(F) This leads to the collapse of the state ψ(F) into a specific eigenstate χn.
ApplyingO(F)on the state vector collapses it to oneO(F)’s eigenstates.
The projection of the state vectorψ to one of the eigenstatesχ n of the operator
In quantum mechanics, the phenomenon known as the collapse of the state vector ψ is both discontinuous and instantaneous This collapse is triggered by the act of measurement, which is a fundamental aspect of quantum theory that operates independently of the Schrödinger equation.
In contrast to classical mechanics, which produces consistent outcomes from identical initial conditions, quantum mechanics introduces inherent uncertainty, resulting in a variety of possible final states For instance, even when radioactive atoms are prepared identically, their decay occurs randomly over time, aligning with the probabilistic nature of quantum mechanics.
After many repeated observations performed on state ψ(F), all of which in principle are identical to each other, the experiment yields the average value of the physical operatorO(F), namely,
The inability of the Schrödinger equation to model the measurement process has sparked ongoing debate among physicists Many believe that the fundamental equations of quantum mechanics should govern not only the evolution of quantum states but also their collapse during measurement.
7 Except, as mentioned earlier, for eigenstates. caused by the process of measurement As of now, there has been no resolution to this conundrum.
Summary: Quantum Entity
In the context of quantum mechanics, a quantum entity is fundamentally indeterminate, challenging our classical understanding of nature A thorough examination of the concepts of entity, thing, and object reveals that describing a quantum entity necessitates a framework that diverges from traditional perspectives.
The foundation of a quantum entity lies in its degree of freedom (F), which is intrinsically indeterminate, leading to quantum indeterminacy Max Born made a significant contribution by proposing that this indeterminacy can be represented by a state vector ψ(F) that follows the principles of quantum probability This state vector is fundamentally linked to the degree of freedom, encapsulating all the information derived from its indeterminate nature, as depicted in Fig 2.7.
The state vector ψ(F) does not directly encompass the degree of freedom in physical space; instead, all observations related to the degree of freedom interact solely with the state vector This means that no observation can directly engage with the degree of freedom itself, as any interaction between the measuring device and the degree of freedom is always mediated by the state vector.
In summary, the following is a definition of the quantum entity:
A quantum entity consists of two essential components: the degrees of freedom (F) and the state vector ψ(F), which encapsulates all its properties This inseparable relationship between the degrees of freedom and the state vector defines the fundamental condition of the quantum entity's existence.
The Copenhagen quantum postulate states that observations collapse a quantum state into a definite eigenstate of a physical operator, resulting in uncertainty for all experimental outcomes except those involving eigenstates Consequently, repeated observations provide measurable quantities that characterize the observable properties of the quantum entity.
Fig 2.7 A quantum entity is constituted by its degrees of freedom F and the state vector ψ( F ) that permanently encompasses and envelopes its degrees of freedom
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Heisenberg developed a foundational framework for quantum mechanics utilizing operator algebra, which defines quantum indeterminacy and quantum probability concepts For a detailed discussion, refer to Chapter 7.
An independent formulation of quantum mechanics, known as the Dirac–Feynman framework, introduces the concept of indeterminate paths and time evolution This formulation is articulated through the Feynman path integral, which is further explored in Chapter 11.
Quantum indeterminacy, initially articulated by Born through the state vector, gained clarity in Heisenberg's operator formulation, where physical quantities are represented by operators tailored for specific measurements The quantum state is encapsulated by the density matrix operator, as elaborated in Chapter 6 These operators extract comprehensive information about the quantum entity by interacting with its quantum state, which contains all observable properties of the quantum system.
Ultimately, the results of an experiment can only reflect the impact of the quantum entity on our measuring devices, often observed as distinct changes or 'clicks' in the device's counters.
Our understanding of physical entities relies heavily on theoretical and mathematical concepts that are grounded in extensive experimental data In quantum mechanics, these mathematical frameworks have allowed us to deduce the existence of quantum entities The theoretical constructs of quantum mechanics are not arbitrary; rather, they are intricately linked to empirical properties, making it improbable that significant gaps or redundancies exist within the theoretical framework of the field.
The founders of quantum mechanics recognized its paradoxical nature, which defies everyday intuition rooted in macroscopic experiences Niels Bohr emphasized this complexity, stating, “Those who are not shocked when they first come across quantum mechanics cannot possibly have understood it.” Similarly, Richard Feynman remarked that “no one understands quantum mechanics,” warning against the futile quest for conventional explanations, as it leads to a dead end where no one has found clarity.
The mysteries and paradoxes of quantum mechanics arise due to the following two reasons:
• The intrinsic indeterminacy and uncertainty exhibited by a quantum entity is completely absent in our everyday life.
• The linearity of the Schrửdinger equation that determines the dynamics of the quantum state.
The mystery of quantum mechanics is not just only about indeterminism but rather also about the manner in which this indeterminism is realized.
Dirac famously stated that a metaphor in scientific theory need not be a classical construct but should illuminate the self-consistency of fundamental laws In this context, the term "trans-empirical" is proposed to enhance the conceptual framework of quantum mechanics This new concept aims to provide a clearer metaphor for quantum equations, with the hope of making the theory more transparent and accessible for deeper mathematical analysis.
The following are the topics covered in this chapter:
• Definition of the concept of trans-empirical.
• The reexamination of the various domains of the quantum entity’s superstructure.
• The study of the two-slit experiment as an exemplar for studying the relation of the trans-empirical domain to quantum superposition.
• The trans-empirical quantum principle is stated and essentially entails enhancing the concept of what “exists” in Nature.
Real Versus Exist
Dirac asserted that quantum mechanics is constructed from physical concepts that defy verbal explanation Niels Bohr echoed this sentiment, suggesting that our reliance on language limits our ability to define reality, leaving us in a state of uncertainty about fundamental concepts such as "up" and "down."
‘reality’ is also a word, a word which we must learn to use correctly” [30].
The terminology used in quantum mechanics is often opaque and complex, necessitating precise definitions for effective discussion and understanding Two fundamental concepts that require careful consideration are the definitions of "reality" and "existence" within the framework of quantum mechanics.
In quantum mechanics, the concepts of "real" and "exists" carry significant implications, making it essential to define their meanings clearly The interpretation of whether something is considered "real" or "exists" hinges on the definitions of these terms.
“exist” is extended so that it can be used to discuss many concepts that arise in quantum mechanics.
According to the Oxford dictionary, "real" refers to something that actually exists or occurs in fact, rather than being imagined or supposed Similarly, "exist" signifies having objective reality or being Therefore, the phrases "to exist" and "to be real" appear to convey similar meanings.
“to exist” meaning “to be real” and “to be real” meaning “to exist”; in fact, these two words are normally used synonymously.
A real entity is defined by its empirical, objective, and factual existence Quantum mechanics embraces this definition of "real" while introducing a new perspective on the nature of existence.
Heisenberg posits that "reality" is defined not by the electron itself, but by the observations made on it, emphasizing that the results of these observations hold empirical significance He interprets "real" as an empirical reality with objective existence, aligning with its dictionary definition For clarity, this book adopts Heisenberg's definition, meaning that "real" will exclusively refer to the outcomes of observations, or what can be deemed empirical.
In quantum mechanics, a particle such as an electron is inherently indeterminate, meaning its position lacks objective reality until measured According to Heisenberg, "Reality is not in the electron," highlighting that what we perceive as real is solely a result of observation While quantum entities do not possess an objective reality like classical entities, they still maintain a mode of existence, as their quantum state persists even when unobserved.
“being” of a quantum entity when it is not being observed, when it is intrinsically indeterminate?
The term "exist" applies to both empirical and trans-empirical entities, emphasizing the concept of objective existence While traditional definitions of "exist" may imply a need for empirical reality, many entities like mathematics and language lack objective existence yet are still recognized as existing This book redefines "exist" to encompass these non-empirical entities, separating the notion of existence from the requirement of objective reality.
The term "real" refers to entities that possess being, which can encompass a broader interpretation beyond just objective existence or reality.
The term "exist" now encompasses entities without objective existence, including those that are inherently indeterminate This updated definition allows for the acknowledgment of unobservable and indeterminate quantum degrees of freedom as existing, highlighting a broader understanding of existence in the context of quantum mechanics.
This book adopts the perspective that the unobservable and indeterminate quantum degree of freedom is a real entity, while also affirming that its empirically observed properties are tangible and exist in reality.
Empirical, Trans-empirical, and Indeterminate
This book delves into the intriguing and puzzling aspect of quantum mechanics, specifically the concept that the degree of freedom, denoted as F, cannot be empirically observed To clarify and comprehend the unique characteristics of this degree of freedom, the term "trans-empirical" is introduced to describe F.
According to the Oxford dictionary, "empirical" refers to knowledge that is derived from observation or experience rather than theoretical or logical reasoning This concept emphasizes the importance of direct experimental observations in understanding the empirical realm.
The term "trans-empirical" refers to a realm of existence that transcends the empirical world This domain is not accessible through direct observation or experiential knowledge, but can only be understood through theoretical frameworks or pure logic.
The term "trans-empirical" refers to the virtual and unobservable existence of a quantum entity, highlighting its distinction from empirically real existence This adjective emphasizes the unique nature of trans-empirical existence, setting it apart from what can be observed and measured in the empirical realm.
A determinate entity exists in a specific state, with this definiteness being an intrinsic property of the entity itself Therefore, a determinate entity is synonymous with an empirical entity, as its quality of determinateness remains inherent regardless of observation.
The term "indeterminate," as defined by the Cambridge Dictionary, refers to something that is not measured, counted, or clearly described However, this definition is overly literal and restrictive, suggesting an objective existence of the indeterminate entity, which this book will not endorse.
The term "indeterminate" is often associated with quantum uncertainty, reflecting an intuitive contrast to concepts that are determinate, definite, and factual To effectively relate "indeterminateness" to quantitative inquiries, it is essential to define the concept more precisely, particularly in the context of experiments, to clarify what can be classified as an "indeterminate" entity.
A classical entity is characterized by having precise and definite values for its dynamical variables at any given moment, making it intrinsically determinate In contrast, a quantum entity is inherently indeterminate, as its quantum state lacks exact and objective existence until a measurement is conducted This distinction highlights the contrasting nature of classical and quantum entities, with the terms determinate and indeterminate effectively capturing their fundamental differences.
The term "indeterminate" is intrinsically linked to the notion of trans-empirical entities, suggesting that indeterminacy describes an entity that lacks objective existence rather than merely a deficiency in precise knowledge In this context, "indeterminateness" is defined as a form of existence that is not observable and transcends empirical understanding.
The words indeterminate and trans-empirical will be used interchangeably, with a specific choice being made depending on the context of the discussion.
In quantum mechanics, the position of a particle is fundamentally indeterminate, meaning it does not occupy a specific location Instead, a quantum particle is present at all points around an average position, highlighting the inherent uncertainty in its spatial coordinates.
The term "virtual" is now avoided due to its association with software simulations like "virtual reality" and "virtual machines," which are unrelated to the indeterminate states of quantum mechanics Quantum particles exist in a trans-empirical state, as they cannot be empirically observed at more than one location simultaneously.
The indeterminate paths discussed in Sect 2.10 and illustrated in Fig 2.6 highlight the concept of quantum particles existing in multiple trajectories simultaneously While there are N determinate paths with specific trajectories, the indeterminate nature of a particle implies that it occupies all allowable paths at once This phenomenon necessitates the existence of the quantum particle in a trans-empirical form, as it cannot be observed at more than one location simultaneously.
In conclusion, the quantum entity’s indeterminacy and its trans-empirical state are inseparable since to be indeterminate the quantum entity must exist in a trans- empirical state and vice versa.
Quantum Mechanics and the Trans-empirical
This book is written within the framework of the Copenhagen interpretation of quantum mechanics, developed primarily by Bohr and Heisenberg [17].
In the Copenhagen interpretation of quantum mechanics, only the quantities that can be experimentally observed are considered real However, this interpretation leaves several questions unanswered, particularly regarding the ontological status of a quantum system when it is not being observed.
This book primarily aims to clarify the ontological status of quantum degrees of freedom when unobserved, as discussed in Section 3.1 It explores the continued existence of quantum entities, emphasizing their state both before and after observation.
The Copenhagen interpretation is expanded by redefining existence in quantum mechanics, proposing that the degree of freedom and quantum state represent real entities rather than mere mathematical constructs This exploration seeks to understand how quantum entities can be considered to exist, ultimately replacing the classical mechanics notion of "objective reality."
The concept of a trans-empirical domain of Nature is introduced to expand the classical understanding of existence beyond the empirical realm This new perspective aims to elucidate the complex phenomena uncovered by quantum mechanics, providing a clearer framework for interpreting its principles and workings.
Classical physics operates solely within the realm of objective phenomena, which are empirically observable In contrast, quantum theory has expanded our understanding of Nature by introducing the concept of a trans-empirical domain, revealing aspects of reality that transcend direct observation.
Trans-empirical Transition Measure- Empirical ment
Degree of freedom State space Dynamics Operators Observation
Fig 3.1 The organization of the superstructure of a quantum entity into the empirical, the trans- empirical, and the transitional domains (published with permission of © Belal E Baaquie 2012. All Rights Reserved)
Physics is fundamentally an empirical science, with all its laws grounded in empirical evidence The relationship between the trans-empirical aspects of Nature and the empirical domain of physics is explored in Section 3.10.
Quantum Superstructure and the Trans-empirical
The five fundamental pillars of the quantum entity's superstructure can be categorized based on empirical and trans-empirical criteria In the following sections, each pillar of quantum mechanics will be examined in terms of what is empirically observable and what is not The findings from this analysis are presented in Figure 3.1.
• The foundation of the quantum entity, namely, its degree of freedom F, is a trans-empirical quantity.
Experimental readings reveal the empirical manifestation of quantum entities, yielding measurable values for the expectation of observable operators, denoted as E V [O(F)].
The quantum state vector serves as a transitional link between the trans-empirical and empirical domains It encapsulates all observable properties of the degree of freedom F, while simultaneously being in contact with it When measurements are made on F, the quantum state collapses into its empirical form, leading to observable results after repeated measurements.
Operators O(F) that represent physical observables facilitate the transition of quantum states from the trans-empirical to the empirical domain While these operators do not directly interact with the degree of freedom F, the state vector ψ(t, F) serves as a mediator, establishing the connection between the operators O(F) and the degree of freedom F.
Fig 3.2 The enveloping theoretical superstructure that describes the nature of a quantum entity (published with permission of © Belal E.
Figure 3.2 depicts the structure and organization of a quantum entity's superstructure, highlighting that its foundation is based on the degree of freedom (F) This foundational aspect supports the entire quantum superstructure, illustrating the interconnectedness of the diverse domains within the quantum entity.
The distinctive features of a quantum entity reveal a mode of existence that embodies an ontology significantly more intricate than that of classical entities.
Quantum Degree of Freedom F Is Trans-empirical
The quantum degree of freedom represents a quantitative entity that exists simultaneously across all permissible values This concept is fundamentally indeterminate and transcends empirical observation, forming the basis of the degree of freedom space, denoted as F.
The Bell analysis reveals that a quantum degree of freedom, when subjected to experimental operators, lacks a precise and determinate value prior to observation, as detailed in Chapter 7.
Before a measurement is conducted, the degree of freedom, denoted as F, exists in a unique mode This mode implies that F does not have a specific value prior to measurement; instead, it encompasses all potential values organized within a geometrical space Metaphorically, the degree of freedom exists across the entire spectrum of its possible values simultaneously However, this mode of existence is not empirically observable, leading to the classification of the degree of freedom as trans-empirical.
The degree of freedom can be understood as an entire space, making it inaccurate to expect it to have a single value; this is akin to attempting to define a whole space by merely providing the coordinates of one point within it This concept illustrates why the degree of freedom lacks a specific value until it is observed.
2 And continues to exist after a measurement as well.
The degree of freedom F is obscured by the state space V(F), preventing direct observation by operators O(F) To gather information about F, the operator O(F) performs a measurement, depicted as a "net" cast around the quantum entity Continuous observations lead to the expectation value E_V[O(F)].
Quantum probability theory associates observables with operators rather than specific values of degrees of freedom, distinguishing it from classical probability theory, where observables are defined as the specific values of a random variable that can be directly measured during sampling.
Operators interact with the quantum state of a quantum entity, with the expectation value of the operator being influenced by the specific state vector Essentially, applying the operator to a degree of freedom indirectly affects the operator itself.
In quantum mechanics, the properties of the degree of freedom (F) remain indeterminate and trans-empirical, as they are obscured by the quantum state vector ψ(F) While empirical quantities, such as the expectation value of physical operators, are determinate and possess unique values, the degree of freedom F is concealed and cannot be directly observed Unlike classical random variables, which can be objectively measured, the specific points within the space of F are masked by the quantum state The quantum entity is defined by its degree of freedom and its state vector, with quantum operators O(F) acting on ψ(F) to indirectly reveal the existence of F This interplay between operators and the quantum state illustrates how the quantum state effectively shields F from direct observation.
The degree of freedom exists as a trans-empirical entity, represented by an entire space rather than a specific value This indeterminate nature of the degree of freedom emphasizes that it cannot be confined to a singular point, but rather encompasses a broader spatial dimension.
F, and hence, it (F) cannot be described by a quantity having a single, determinate and fixed value.