2.2 NEED FOR QUANTUM DESCRIPTION
2.2.1 Some experiments that ushered in the quantum age
Classical physics has proven to be adequate to describe most physical phenomena. It is successful in describing planetary motion, trajectories of particles, wave propagation, etc. At the beginning of the twentieth century, some observations were made which started shaking the foundations of classical thinking. In this section we will examine some of the critical experiments that eventually led to quantum mechanics. In this text we will focus only on non-relativistic quantum mechanics. This means that we will consider situations where the speed of particles (other than photons) is much slower than the speed of light.
Waves behaving as particles: blackbody radiation
An extremely important experimental discovery which played a central role in the devel- opment of quantum mechanics was the problem of the spectral density of a blackbody radiation. If we take a body with a surface that absorbs any radiation (a blackbody) we find that it emits radiation at different wavelengths. The intensity (power per unit area) of the radiation emitted between wavelengths A and A + dX is defined as
dI = R(X)dX (2.1) where R(X) is called the radiancy. The spectral dependence of R(X) is found to have a certain dependence on the wavelength and temperature of the blackbody. In Fig. 2.1 we show how the experimentally observed R(X) behaves at different temperatures. Several interesting experimental observations are made in regard to the emitted radiation:
The total intensity has the behavior
1= R(X)dXocT4 Jo
or
/ = (TT4 (2.2)
This is known as Stefan's law. The constant <r is called the Stefan-Boltzmann constant.
It is found to have a value
(j = 5.67x 10"8 Wm"2K"4
• The radiancy versus wavelength plot shows that there is a maximum at a certain
2.2. Need for quantum description 41 wavelength Amax as can be seen from Fig. 2.1. The temperature dependence of this wavelength is given by
The proportionality constant is given by the relation
Am a xT = 2.898 x 10"3 meter-Kelvin This relation is known as Wien's displacement law.
Using classical physics a formalism has been developed to understand blackbody radiation. According to classical physics, radiancy is given by the Rayleigh-Jeans law,
R(\) = %kBTC- (2.4)
However, when careful experiments were carried out and the spectral density tabulated, it was found that the Rayleigh-Jeans law was applicable only in a small frequency range. In fact, as can be seen from the equation, the classical law predicts an infinite energy density at very short wavelengths - an obviously unphysical result. It can be seen from Fig. 2.1 that while the classical law gives a reasonable fit to experiments for long wavelengths, it completely fails at short wavelengths. The entire spectrum was only understood when Planck suggested that an electromagnetic wave with frequency UJ exchanges energy with matter in a "quantum" given by
E = hv = hu) (2.5) Here h is a universal constant called Planck's constant. The symbol h stands for /Z/2TT.
This assumption seemed to suggest that light waves have a well-defined energy just as particles do.
The quantity h or h that has been introduced is called Planck's constant and has a value
h = 6.6261 x 10"3 4 J - s
h = ^ - = 1.05 x 1(T34 J -s (2.6) Waves behaving as particles: photoelectric effect
It is well known that electromagnetic waves are described by Maxwell's equations. Phe- nomena such as interference and diffraction are well explained by Maxwell's equations.
An important outcome of the wave theory is that the energy of a light beam can change continuously. As the intensity of the wave increases, the energy carried by a light beam increases. This seems quite intuitive and in most experiments this expectation is indeed verified. However, in 1887 Heinrich Hertz carried out an experiment which the wave theory of light was unable to explain. The experiment is known as the photoelectric effect and was the basis for Einstein's model for how light behaves.
In the photoelectric experiment light falls upon a material system and electrons are knocked out due to the interaction of the light with electrons. A typical experimental
<
0.3
0.2
0.1
visible region ENERGY DISTRIBUTION IN A BLACKBODY RADIATION Planck introduced the light quanta of energy hv to explain the energy
distribution.
5,000 10,000 15,000 WAVELENGTH IN ANGSTROMS
20,000
Figure 2.1: Measured spectral energy distribution of a blackbody radiation. The explanation of such experimental observations forced Planck to introduce a constant h (now called Planck's constant).
arrangement used is shown in Fig. 2.2. A potential is applied between the emitter and the collector. If the impinging light cannot knock an electron out of the metal emitter, there will be no photocurrent. If electrons are knocked out these electrons can make it to the collector if their energy is larger than the potential energy eVext between the emitter and the collector.
Let us assume that the electrons in the metal need to overcome an energy e<j>
in order to escape from the metal. The quantity e<f> is called the work function of the metal and arises from the binding of the electrons to the metal ion. If the impinging light beam gives the emitted electrons an energy Eem, the electrons will emerge from the metal with an energy Eem — e<f>. An opposing bias is applied to the emitter-collector and the value of this bias is adjusted so that the electrons emitted are just unable to make it to the collector, i.e., the photocurrent becomes zero. This value of the applied voltage Vs is called the stopping voltage. The current will go to zero when
- e S - eVs (2.7)
Experimentally we can measure Vs as a function of the intensity and frequency of light.
Classical wave theory suggests that the following observations should be made in the photoelectric effect:
• The energy with which the electrons should emerge from the metal should be propor- tional to the intensity of the light beam. Thus, as the intensity increases, the stopping voltage should also increase.
• The electron emission should occur at any frequency provided the intensity of the light beam is sufficiently high.
2.2. Need for quantum description 43
• There should be a time interval At between the switching on of the light beam and the emission of electrons. If A is the area over which the electron is confined (roughly equal to the area of an atom or 10"19 m2), the time it should take the electron to gain an energy AE is
A, A E
where / is the light intensity. If we use AE ~ 1.0 eV and / ~ 1 W/cm2, we find that A T - 10"3 s.
The three expectations from classical physics all seem consistent with our phys- ical intuition. However, actual experiments show them to be incorrect. Instead, the following occurs:
• If the frequency of light is below a cutoff value, there is no emission of electrons, regardless of intensity as shown in Fig. 2.2b.
• The stopping potential is completely independent of the intensity of light. A typical result is shown in Fig. 2.2c. As can be seen from this figure, the stopping voltage is unaffected by intensity, although the photocurrent scales with intensity.
• The initial electrons are emitted within a nanosecond or so of the light being turned on. There is essentially no delay between the impingement of light and electron emission.
The experimental observations were thus completely opposed to what was ex- pected on the basis of the wave theory for electromagnetic radiation. It was clear that a radical new interpretation of light was needed. As noted in the previous subsection, Max Planck had developed his formalism to explain the spectral density of blackbody radiation. Based on Planck's ideas, Einstein saw that the photoelectric effect could be explained if light was regarded as made up of particles with energy
E = hco = hv (2.8) Thus light was to be regarded not as waves but discrete bundles or quanta of energy.
These quanta were later called photons.
In Einstein's theory electrons are emitted by a single photon knocking the electron out. Thus the kinetic energy of the emitted electron is
Ee(K.E) = hv-e</> = eVs (2.9) There is no dependence of the electron energy on the intensity of light. A beam with higher intensity has more photons, but each photon has the same energy. The cutoff frequency for electron emission is given by the relation
h v = e<f> (2.10) In Fig. 2.3 we show the work function values for several metals. Also shown are the dependence of stopping voltage on the frequency of light. The slope of this curve is h/e.
Particles behaving as waves: atomic spectra
An area where experiments baffled classical physics was atomic spectra and properties
Light
Emitter Collector
Intensity I2 > I\
Intensity Ix FREQUENCY OF LIGHT (V)
(b)
(a) Einstein explained the
photoelectric effect by treating light as composed
of energy quantas with
energy hv.
v_y
0
POTENTIAL DIFFERENCE V
(c)
Figure 2.2: (a) A schematic of the experimental setup used for studying the photoelectric effect; (b) photocurrent as a function of the frequency of the impinging light for a fixed applied bias; and (c) photocurrent versus applied bias for when the optical signal frequency is above the threshold frequency.
£ 3
•z.
UJ O
Q_
CD 2 Q_
OQ_
CO
60 80 100 120
RADIATION FREQUENCY (101 3 H Z )
Figure 2.3: Stopping voltage versus frequency results for sodium. The slope of the curve is h/e. Also shown are the work functions of several metals.
Material Na
Al Co Cu Zn Ag Pt Pb
e<|> (eV) 2.28 4.08 3.90 4.70 4.31 4.73 6.35 4.14
2.2. Need for quantum description 45 Electrons
Z: Electrons R: atomic radius
Figure 2.4: Thomson model for an atom. The Z electron with charge — e are point particles embedded in a uniformly charged positive sphere of radius R.
of atoms. The question of relevance is: What are the electron energies and trajectories in an atom? One of the earliest models for the atom was proposed by J. J. Thomson, who was the first to identify the electron and measure the ratio of its charge to mass.
Thomson built the atomic model on the basis of classical physics. He assumed that the atom was made up of a uniform sphere with positive charge Ze and radius R in which negatively charged electrons were embedded, as shown in Fig. 2.4. The size of the positively charged sphere is assumed to be of the order of an Angstrom or so and the electrons are assumed to be embedded in the uniform charge at various distances.
If r is the distance of an electron from the center, the force on it is (from classical electrostatics)
Ze2r
F = - — (2.11)
A R 3 V }
At equilibrium there would be a balancing force from the other negatively charged electrons. Away from the equilibrium position there is a linear restoring force on the electron and it is assumed the electron will oscillate about its mean position just as a pendulum does.
The frequency of the oscillation is (according to classical physics)
v — 1
2^ (2.12)
where ra0 is the electron mass. The oscillating electron would radiate electromagnetic radiation of frequency v.
Experiments showed that the frequencies of radiation emitted from atoms were not in agreement with what the model predicted. Also, scattering experiments, in which scattering of alpha particles from atoms was studied, showed that most of the atom was empty. It was found that the positive charge was not distributed over an Angstrom but over a much smaller region.
Advances in optical spectroscopic techniques made possible direct measure- ments of the frequencies of the emitted and absorbed radiation by atoms. When atoms are excited (say, by electromagnetic radiation), they can absorb the radiation. Once they absorb radiation, they can emit radiation as well. It was found experimentally that emitted and absorbed spectra from a species of atoms consisted of several series of sharp lines; i.e., discrete frequencies. It was possible to fit simple relations to the posi- tions of these lines. For example, Johannes Balmer found that the emission wavelengths of hydrogen in the visible regime could be fitted to the relation (the Balmer formula)
2
An= 364.5-^ - n m ; n = 3 , 4 , . . . (2.13) n2 — 4
In fact, other groups of lines in H-spectra were fitted to other expressions. For example, we have the following sequences of optical wavelengths:
Paschen Series:
n2
\n = 820.1— -2 nm; n = 4 , 5 , . . . (2.14) Lyman Series:
2
An = 9 1 . 3 5 - ^ — n m ; n = 2 , 3 , . . . (2.15) n2 — 1
Other atoms were found to have spectra which satisfied similar relations. In Fig. 2.5 we show series of lines observed in atomic spectra of hydrogen.
The observation of atomic spectra showed that for some reason electrons inside an atom can only have certain well-defined energies-not a continuum of energies. With- out fully explaining why this should occur, Bohr came up with a model that explained the results shown in Fig. 2.5. Bohr assumed that the nucleus was essentially a point particle and the electrons spun around the nucleus, just as planets orbit the sun. How- ever, unlike the planets, the electrons can only go around in orbits in which the angular momentum was an integral multiple of h. If r is the radius of an orbit, we must have
rriQvr — n h ; n = 1, 2 . . . (2.16) By proposing this postulate, Bohr made a daring leap. He was able to fit the emission and absorption spectra of the hydrogen atom and he was able to explain why the electron does not radiate continuously, even though it is orbiting the nucleus. The electron cannot radiate electromagnetic energy unless it jumps from one allowed orbit to another.
Based on the postulate that the electron orbits were quantized Bohr was able to calculate the allowed energies of electrons in an atom. Equating the centripetal force and the Coulombic force we get
(2.17)
4TT6O r2
The kinetic energy is now
1 1 e2
K=-m0v2 = - (2.18)
2 8TT60 r
2.2. Need for quantum description 47
90 nm 100 nm 110 nm 120 nm
Lyman (ultraviolet)
ABSORPTION LINES
400 nm 500 nm 600 nm
Balmer (visible)
EMISSION LINES
0.5 nm 1.0 nm 1.5 nm 2.0 nm
Paschen (infrared)
EMISSION LINES
1.0 jxm 2.0 [im 3.0 |im 4.0 ^m
Brackett (infrared)
EMISSION LINES
2.0 nm 4.0 nm
i i
6.0 nnn
i
8.0 nm
Pfund (infrared)
EMISSION LINES
j
Figure 2.5: Emission and absorption lines in the hydrogen atom spectra. There is a regularity in the spacings of the spectral lines and the lines get closer as they reach the upper limit of each series (denoted by the dashed lines).
The potential energy is the Coulombic energy
L (2.19)
47T60 r
and the total energy is
E = -^—- (2.20)
87T60 V
From Eqs. 2.16 and 2.17, we have
e2 e2 1
(mov)v = o r i ; = — (2.21) From this equation and Eq. 2.16, we have, for the allowed orbit radii
rn = iÊl(ằJi)> (2.22)
m0e2
Substituting this equation into Eq. 2.20 for the electron energy, we have moe4 1 t , 9
= - ^ e V (2-23)
The allowed energy levels are shown in Fig. 2.6. The allowed radii of the orbits are, from Eq. 2.22
(2.24) raoe
where
ao = 4 ^ = 0 > 5 2 9 A ( 2 > 2 5 )
Based on his model Bohr was able to provide a model which was consistent with ob- servations made on the hydrogen atom. For example, the emission lines resulted from an electron jumping from a higher energy level to a lower energy level, as shown in Fig. 2.8b. Absorption lines resulted from reverse transitions. The Bohr model, although pioneering, was not found to be adequate to describe the spectra of other atoms. It also failed to explain many other experiments.