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Case Study 6
Quantisation and Quanta after 1905
Case Study 6
Quantisation and Quanta after 1905
In Case Study 6,we took the story of quantisation and quanta up to the publication of
Einstein’s great paper of 1905. To recapitulate the story so far:
• Planck derived the Planck formula for black-body radiation in 1900 using dodgy
arguments which no-one could understand. Planck had quantised the oscillators
and regarded this as no more than a “formal procedure”.
• There were no papers on quanta until Einstein’s paper of 1905.
• Einstein quantised the radiation field rather than the oscillators which are in
equilibrium with the radiation.
• Among the predictions of the theory was an explanation for the photoelectric effect.
Einstein’s Derivation of Planck’s Law (1)
In 1906, Einstein realised that he could reformulate Planck’s ideas using Boltzmann’s
statistical mechanics properly. The system of oscillators can take only discrete energy
levels with energies, 0, ǫ, 2ǫ, 3ǫ, . . . . Boltzmann’s expression for the probability that
an oscillator is in a state with energy E = nǫ in thermal equilibrium at temperature T
where n is an integer is
p(E) ∝ e−E/kT .
Suppose there are N0 oscillators in the ground state, E = 0. Then, the number in the
n = 1 level is N0 e−ǫ/kT , the number in the n = 2 level is N0 e−2ǫ/kT , and so on.
Therefore, the average energy of the oscillator is found in the usual way:
E =
=
N0 · 0 + ǫ · N0e−ǫ/kT + 2ǫ · N0e−2ǫ/kT + . . .
N0 + N0e−ǫ/kT + N0e−2ǫ/kT + . . .
,
N0ǫ e−ǫ/kT [1 + 2(e−ǫ/kT ) + 3(e−ǫ/kT )2 + . . .
N0[1 + (e−ǫ/kT ) + (e−ǫ/kT )2 + . . . ]
.
Einstein’s Derivation of Planck’s Law (2)
We recall the following series expansions:
1
= 1 + x + x2 + x3 + . . .
(1 − x)
1
2 + ...
=
1
+
2x
+
3x
(1 − x)2
Hence, setting x = e−ǫ/kT , the mean energy of the oscillator is
Ē =
ǫ e−ǫ/kT
ǫ
=
1 − e−ǫ/kT
eǫ/kT − 1
If we now follow the classical Boltzmann procedure, we allow the energy elements ǫ to
tend to zero and then, expanding eǫ/kT − 1 for small values of ǫ,
2
3
1
ǫ
ǫ
1
ǫ
+
+
+ ··· − 1
eǫ/kT − 1 = 1 +
kT
2! kT
3! kT
Einstein’s Derivation of Planck’s Law (3)
Thus, for small values of ǫ/kT ,
e
ǫ/kT
ǫ
−1=
kT
and so
ǫ
ǫ
=
= kT
E = ǫ/kT
ǫ/kT
e
−1
Thus, if we take the correct classical limit, we recover Maxwell’s equipartition theorem
for an oscillator, namely, the average energy of an oscillator in thermal equilibrium is
E = kT .
In order to obtain Planck’s law, however, we must not allow ǫ to go to zero. According to
this simple model, the only allowable energy states are those corresponding to
0, ǫ, 2ǫ, 3ǫ, . . . . Einstein recognised immediately the relation between this proper
formulation of the quantisation of the oscillators and his previous analysis of the
quantisation of the radiation itself.
Einstein’s Derivation of Planck’s Law (4)
In his own words,
‘We must assume that, for ions which can vibrate at a definite frequency and
which make possible the exchange of energy between radiation and matter,
the manifold of possible states must be narrower than it is for bodies in our
direct experience. We must in fact assume that the mechanism of energy
transfer is such that the energy can assume only the values 0, ǫ, 2ǫ, 3ǫ, . . . ’
Thus, the expression for the energy density of radiation per unit frequency interval
becomes:
8πν 2
ǫ
8πhν 3
8πν 2
E=
=
,
u(ν) =
hν/kT
3
c3
c3 eǫ/kT − 1
c (e
− 1)
if we identify the energy elements with the energy of the quanta of light ǫ = hν.
Einstein’s Theory of Specific Heats
In the same paper, Einstein also addressed the problem of the specific heats of solids
at low temperatures. At room temperature, the specific heat of many solids is described
by Dulong and Petit’s Law according to which the heat capacity per mole of a solid is
roughly 3R where R is molar gas constant, R = 8.3145 J k−1 mol−1.
We suppose that the solid consists of N0 atoms per mole and that they all vibrate in
three independent directions. Since, in equilibrium, each mode of vibration has energy
kT , the total internal vibrational energy of the solid is 3N0kT . Therefore, the total
internal energy is U = 3N0kT and so the heat capacity is
C=
dU
= 3N0k = 3R
dT
Einstein’s Theory of Specific Heats
Some solids, such as beryllium, boron and carbon, had smaller heat capacities than
3R. In addition, the specific heats of solids decrease with decreasing temperature, 3R
only being attained at high temperatures.
Einstein proposed that his quantum formula for the average energy of an oscillator at
temperature T should be used,
ǫ
E = ǫ/kT
e
−1
rather than the classical formula. For simplicity, he assumed that solids vibrate at a
frequency νE, known as the Einstein frequency. Then, the internal energy of the solid is
hν
.
U = 3N0E = 3N0 hν /kTE
e E
−1
The heat capacity is found by differentiation
dU
hνE 2
ehνE/kT
C=
= 3R
.
hν
/kT
2
dT
kT
(e E
− 1)
Comparison of Einstein’s Theory with
Measurements of the Heat Capacity of Diamond
In his paper of 1906/7, Einstein
compared the theory with
measurements of the heat capacity
of diamond and found encouraging,
if not perfect, agreement.
This was an important prediction
because Walther Nernst was
beginning to measure the heat
capacities of solids at low
temperatures.
Einstein on Fluctuations in Black-body Radiation
One of Einstein’s most beautiful papers on quanta was written in 1909 and concerned
the random fluctuations in black-body radiation. Einstein began by reversing
Boltzmann’s relation between entropy and probability.
W = eS/k
Now divide up the volume V into a large number of cells and suppose that ∆ǫi is the
energy fluctuation in the ith cell. Then, the entropy of the cell is
Si = Si(0) +
∂ 2S
!
∂S
2 + ...
1
(∆ǫ
)
∆ǫi + 2
i
∂E
∂E 2
But, averaged over all cells, we know that there is no net fluctuation,
therefore
S=
X
1
Si = S(0) + 2
∂ 2S
∂E 2
!
X
(∆ǫi )2 + . . .
P
∆ǫi = 0, and
Einstein (1909)
Therefore, the probability distribution of the fluctuations is
"
W = (const) exp 1
2
∂ 2S
∂E 2
!P
(∆ǫi )2
k
#
.
This is the sum over a set of normal distributions and so, for any single cell, we can
write
"
(∆ǫi)2
1
Wi = (const) exp − 2
σ2
#
where
σ2 = k
−∂ 2S/∂E 2
.
We can now go through the usual process of inverting the Planck distribution to express
1/T in terms of the energy density of the radiation and identifying it with (∂S/∂E)
where the energy can be taken to be E = V u(ν) ∆ν.
!
∂S
1
k
8πhν
= =
ln 3
+1 .
∂E
T
hν
c u(ν)
Fluctuations in Black-body Radiation
It is now straight-forward to find σ
σ2 = −
c3
k
2
=
hνE
+
E
(∂ 2S/∂E 2)
8πν 2V ∆ν
!
In terms of the fraction fluctuations, we can write
σ2
=
2
E
c3
hν
+
E
8πν 2V ∆ν
!
This is a remarkable formula. The first term arises from the Wien region of the
spectrum and is the statistical fluctuation expected if there are N = E/hν photons
present, recalling that the fractional fluctuation is expected to be ∆N/N ≈ 1/N 1/2.
Fluctuations in Black-body Radiation
The second term arises from the Rayleigh-Jeans region of the spectrum. It represents
the fluctuations due to the random superposition of waves. As was shown earlier, the
number of modes in the frequency range ν to ν + ∆ν is
8πν 2 ∆νV
.
Nmode =
c3
When we superimpose waves of a single mode with random phases, the fluctuations in
the energy correspond to (∆E/E)2 = 1 (see TCP2, Chapter 15, if necessary) and so
the second term tells us that
c3
1
∆E 2
=
.
=
2
2
E
Nmode
8πν ∆νV
I find this an amazing result. It says that we should add together statistically the wave
and particle aspects of the radiation field to find the total fluctuation in the radiation
intensity.
The 1911 Solvay Conference
In 1910, Nernst found good
agreement between his
measurements of the heat capacity
of materials at low temperatures
and Einstein’s theory of the specific
heats of solids. He then persuaded
the Belgian industrialist Solvay to
sponsor the First Solvay
Conference in 1911. This meeting
made the issues concerning of
quantum physics widely known in
the physics community.
The Reception of the Concepts of Quanta
Although the community in general
found the ideas of quanta hard to
swallow, by the 1911 Solvay
conference, the mood was
swinging in favour of quanta.
Those in favour included Lorentz,
Nernst, Planck, Rubens,
Sommerfeld, Wien, Warburg.
Langevin, Hasenohrl, Onnes.
The solid circles show the number of authors
who published on quanta each year. Open
circles - black body radiation.
Those against included Poincaré
and Jeans.
Rutherford, Brillouin, Marie Curie,
Perrin and Knudsen were neutral.
Bohr’s Theory of the Hydrogen Atom
At this point, we need to change gears and review a number of the other great
discoveries and problems which had arisen in the understanding of atomic physics over
the period 1895 to 1911. The key events of this part of the story were:
• Thomson and the discovery of the Electron.
• Rutherford and the discovery of the nucleus.
• The structure of atoms and the Bohr model of the hydrogen atom.
Thomson and the Discovery of the Electron
The discovery of the electron in 1897 is traditionally attributed to J.J. Thomson. He
showed that the charge-to-mass ratio, e/me, of cathode rays is about two thousand
times that of hydrogen ions. But others were hard on his heels.
• In 1896, Pieter Zeeman discovered the broadening of spectral lines when a sodium
flame is placed between the poles of a strong electromagnet. Lorentz interpreted
this result as the splitting of the spectral lines due to the motion of the ‘ions’ in the
atoms about the magnetic field direction – a lower limit of 1000 was found for the
value of e/me .
The Discovery of the Electron
• In January 1897, Emil Wiechert used the magnetic deflection technique to obtain a
measurement of e/me for cathode rays and concluded that these particles had
mass between 2000 and 4000 times smaller than that of hydrogen, assuming their
electric charge was the same as that of hydrogen ions. He obtained only an upper
limit to the speed of the particles since it was assumed that the kinetic energy of
the cathode rays was Ekin = eV , where V is the accelerating voltage of the
discharge tube.
• Walter Kaufmann’ experiment was similar to Thomson’s. He found the same values
of e/me, no matter which gas filled the discharge tube, a result which puzzled him.
He found a value of e/me 1000 times greater than that of hydrogen ions. He
concluded ‘. . . that the hypothesis of cathode rays as emitted particles is by itself
inadequate for a satisfactory explanation of the regularities I have observed.’
Thomson’s Contributions (1)
J.J. Thomson was the first to interpret the experiments in terms of a sub-atomic particle.
In his words, cathode rays constituted
‘. . . a new state, in which the subdivision of matter is carried very much further
than in the ordinary gaseous state.’
• In 1899, Thomson used one of C.T.R. Wilson’s early cloud chambers to measure
the charge of the electron. He counted the total number of droplets formed and
their total charge. From these, he estimated e = 2.2 × 10−19 C, compared with
the present standard value of 1.602 × 10−19 C. This experiment was the
precursor of the famous Millikan oil drop experiment, in which the water-vapour
droplets were replaced by fine drops of a heavy oil, which did not evaporate during
the course of the experiment.
Thomson’s Contributions (2)
• Thomson also demonstrated that the β-particles emitted in radioactive decays and
those ejected in the photoelectric effect had the same charge-to-mass ratio as the
cathode rays.
• Thomson pursued a much more sustained and detailed campaign than the other
physicists in establishing the universality of what became known as electrons, the
name coined for cathode rays by Johnstone Stoney in 1891.
• It seems fair to regard Thomson as the discoverer of the first sub-atomic particle.
How Many Electrons Are There in the Atom?
• There might have been as many as 2000 electrons, if they were to make up the
mass of the hydrogen atom. The answer was provided by a brilliant series of
experiments carried out by Thomson and his colleagues, who studied the
scattering of X-rays by Thomson scattering in thin films.
• Thomson worked out the scattering rate using the classical expression for the
radiation of an accelerated electron. The cross-section for the scattering of a beam
of incident radiation by an electron is
e4
8πre2
−29
2
=
6.653
×
10
m
,
σT =
=
2
2
4
3
6πǫ0me c
the Thomson cross-section, where re = e2/4πǫ0mec2 is the classical electron
radius.
How Many Electrons Are There in the Atom?
• In collaboration with Charles Barkla, Thomson showed that, except for hydrogen,
the number of electrons was roughly half the atomic weight. The number of
electrons and, consequently, the amount of positive charge in the atom increased
by units of the electronic charge.
• The key question was, ‘How are the electrons and the positive charge distributed
inside atoms?’ In the picture favoured by Thomson, the positive charge was
distributed throughout the atom and, within this sphere, the negatively-charged
electrons were placed on carefully chosen orbits – the rather subtle ‘plum-pudding’
model of the atom (see later).
The Discovery of the Nucleus
The discovery of the nucleus resulted from a brilliant series of experiments carried out
by Rutherford and his colleagues Hans Geiger and Ernest Marsden in the period
1909-12 while Rutherford was at Manchester University. α-particles pass through thin
films rather easily, suggesting that much of the volume of atoms is empty space,
although there was evidence of small-angle scattering. Rutherford persuaded Marsden
to investigate whether α-particles were deflected through large angles on being fired at
a thin gold foil. A very small number of particles almost returned along the direction of
incidence. In Rutherford’s words:
‘It was quite the most incredible event that has ever happened to me in my life.
It was almost as incredible as if you fired a 15-inch shell at a piece of tissue
paper and it came back and hit you.’
Typically, the α-particles were travelling at 10,000 km s−1.
The Discovery of the Atomic Nucleus
In 1911, Rutherford hit upon the idea that, if all the positive charge were concentrated in
a compact nucleus, the scattering could be attributed to the repulsive electrostatic force
between the incoming α-particle and the positive nucleus. He used his knowledge of
central orbits in inverse-square law fields of force to work out the properties of what
became known as Rutherford scattering. The angle of deflection φ is
4πǫ0mα
φ
2
p
v
cot =
0 0,
2
2Ze2
where p0 is the collision parameter, v0 is the initial velocity of the α-particle and Z the
nuclear charge. The probability that the α-particle is scattered through an angle φ is
φ
1
p(φ) ∝ 4 cosec4 ,
2
v0
the famous cosec4(φ/2) law derived by Rutherford, which was found to explain
precisely the observed distribution of scattering angles of the α-particles.
The Discovery of the Atomic Nucleus
The fact that the scattering law was obeyed so precisely, even for large angles of
scattering, meant that the inverse-square law of electrostatic repulsion held good to
very small distances indeed. The nucleus had to have size less than about 10−14 m,
very much less than the sizes of atoms, which are typically about 10−10 m.
The papers by Rutherford, Geiger and Marsden from 1909 to 1913 are classics of 20th
century physics. Rutherford attended the first Solvay Conference in 1911, but made no
mention of his remarkable experiments, which led directly to his nuclear model of the
atom. Remarkably, this key result for understanding the nature of atoms made little
impact upon the physics community at the time, and it was not until 1914 that
Rutherford was thoroughly convinced of the necessity of adopting the nuclear model of
the atom. Before that time, however, someone else did – Niels Bohr, the first theorist to
apply successfully quantum concepts to the structure of atoms.
Problems of Atomic Models
• The electrons in the atom cannot be stationary because of Earnshaw’s theorem,
which states that any static distribution of electric charges is mechanically unstable.
They either collapse and disperse to infinity under the action of electrostatic forces.
• The alternative is to place the electrons in orbits, the ‘Saturnian’ model of the atom.
The most famous of these early models was that due to the Japanese physicist
Nagaoka, who attempted to associate the spectral lines of atoms with small
vibrational perturbations of the electrons about their equilibrium orbits. The
problem with this model was that perturbations in the plane of the electron’s orbit
are unstable, leading to instability of the atom as a whole.
Radiative Instability
Suppose the electron is in a circular orbit of radius a. Equating the centripetal force to
the electrostatic force of attraction between the electron and the nucleus of charge Ze,
mev 2
Ze2
= me|r̈|
=
2
4πǫ0a
a
where |r̈ | is the centripetal acceleration. The time it takes the electron to lose all its
kinetic energy by radiation is
2πa3
E
=
T =
|dE/dt|
σTc
Taking the radius of the atom to be a = 10−10 m, the time it takes the electron to lose
all its energy is about 3 × 10−10 s. Something is profoundly wrong. As the electron
loses energy, it moves into an orbit of smaller radius, loses energy more rapidly and
spirals into the centre.
The Plum-Pudding Model
The solution was to place the electrons in orbits such that there is no net acceleration
when the acceleration vectors of all the electrons in the atom are added together so
that the electrons had to be very well ordered in their orbits. If there are two electrons in
the atom, they can be placed in the same circular orbit on opposite sides of the nucleus
and so, to first order, there is no net dipole moment as observed at infinity, and hence
no dipole radiation. There is, however, a finite electric quadrupole moment and hence
radiation at the level (λ/a)2, relative to the intensity of dipole radiation. Since
λ/a ∼ 10−3, the radiation problem can be significantly relieved. By adding more
electrons to the orbit, the quadrupole moment can be cancelled out as well and so, by
adding sufficient electrons to each orbit, the radiation problem can be reduced to
manageable proportions. This was the basis of Thomson’s plum-pudding model of the
atom.
Niels Bohr
• Niels Bohr completed his doctorate on the
electron theory of metals in 1911. He
convinced himself that this theory was
seriously incomplete and required further
mechanical constraints on the motion of
electrons at the microscopic level.
• He spent the following year in England, working
for 7 months with J.J. Thomson at the
Cavendish Laboratory in Cambridge, and four
months with Ernest Rutherford in Manchester.
Bohr was immediately struck by the
significance of Rutherford’s model of the
nuclear structure of the atom and began to
devote all his energies to understanding atomic
structure on that basis.
Bohr and the Quantum Theory
• He appreciated the distinction between the chemical properties of atoms,
associated with the orbiting electrons, and radioactive processes associated with
activity in the nucleus. On this basis, he could understand the nature of the
isotopes of a particular chemical species.
• Bohr realised that the structure of atoms could not be understood on the basis of
classical physics. The obvious way forward was to incorporate the quantum
concepts of Planck and Einstein into the models of atoms. Recall Einstein’s
statement,
‘. . . for ions which can vibrate with a definite frequency, . . . the manifold of
possible states must be narrower than it is for bodies in our direct
experience.’
This was precisely the type of constraint which Bohr was seeking.
The Earliest Attempts
In 1910, a Viennese doctoral student, A.E. Haas, realised that, if Thomson’s sphere of
positive charge were uniform, an electron would perform simple harmonic motion
through the centre of the sphere. For a hydrogen atom, for which Q = e, the frequency
of oscillation of the electron is
1
ν=
2π
e2
4πǫ0mea3
!1/2
.
Haas argued that the energy of oscillation of the electron, E = e2/4πǫ0a, should be
quantised and set equal to hν. Therefore,
πmee2a
2
.
h =
ǫ0
According to Haas’s approach, Planck’s constant was simply a property of atoms,
whereas those already converted to quanta preferred to believe that h had much
deeper significance.
Bohr’s First Attempt (1)
In the summer of 1912, Bohr wrote an unpublished Memorandum for Rutherford, in
which he made his first attempt at quantising the energy levels of the electrons in
atoms. He proposed relating the kinetic energy T of the electron to the frequency
ν ′ = v/2πa of its orbit about the nucleus through the relation
2 = Kν ′ ,
T =1
m
v
e
2
where K is a constant which he expected would be of the same order of magnitude as
Planck’s constant h.
This criterion fixed the kinetic energy of the electron about the nucleus. For a bound
circular orbit,
Ze2
mv 2
=
a
4πǫ0a2
where Z is the positive charge of the nucleus in units of the charge of the electron e.
Bohr’s First Attempt (2)
The binding energy of the electron is
U
Ze2
2
1
= −T = ,
E = T + U = 2 mev −
4πǫ0a
2
where U is the electrostatic potential energy. The quantisation condition enables both v
and a to be eliminated from the expression for the kinetic energy of the electron, so that
mZ 2e2
T =
.
2
32ǫ2
K
0
which was to prove to be of great significance for Bohr.
John William Nicholson
In 1912, the Cambridge physicist John William Nicholson showed that, although the
Saturnian model of the atom is unstable for perturbations in the plane of the orbit,
perturbations perpendicular to the plane are stable for orbits containing up to five
electrons. The frequencies of the stable oscillations were multiples of the orbital
frequency and he compared these with the frequencies of the lines observed in the
spectra of bright nebulae, particularly with the ‘nebulium’ and ‘coronium’ lines.
Performing the same exercise for ionised atoms with one less orbiting electron, further
matches to the astronomical spectra were obtained. The frequency of the orbiting
electrons remained a free parameter. When he worked out the angular momentum
associated with them, Nicolson found that they were multiples of h/2π. This work
perplexed Bohr.
Bohr (1913)
The breakthrough came in early 1913, when H.M. Hansen told Bohr about the Balmer
formula for the wavelengths, or frequencies, of the spectral lines in the spectrum of
hydrogen,
ν
1
1
1
= =R
− 2 ,
(1)
2
λ
c
2
n
where R = 1.097 × 107 m−1 is the Rydberg constant and n = 3, 4, 5, . . . . As Bohr
recalled much later,
‘As soon as I saw Balmer’s formula, the whole thing was clear to me.’
Bohr could determine the value of his constant K from the running term in 1/n2 in the
Balmer formula which can be associated with the energy of the orbit. For hydrogen with
Z = 1,
mee2
T =
32ǫ0n2K 2
How Bohr Derived the Quantisation Condition
Then, when the electron changes from an orbit with quantum number n to that with
n = 2, the energy of the emitted radiation would be the difference in kinetic energies of
the two states. Applying Einstein’s quantum hypothesis, this energy should be equal to
hν. Bohr found that the constant K was exactly h/2. Therefore, the energy of the state
with quantum number n is
mee2
T =
8ǫ0n2h2
The angular momentum of the state could then be found by writing
′ 2 = 2π 2m a2ν ′ 2 . It immediately follows that
T =1
Iω
e
2
nh
2π
This is how Bohr arrived at the quantisation of angular momentum according to the ‘old’
quantum theory.
J = Iω ′ =
The Pickering Series
In the first paper of his trilogy of 1913, Bohr noted that a similar formula could account
for the Pickering series, which had been discovered in 1896 by Edward Pickering in the
spectra of stars. In 1912, Alfred Fowler discovered the same series in laboratory
experiments. Bohr argued that singly-ionised helium atoms would have exactly the
same spectrum as hydrogen, but the wavelengths of the corresponding lines would be
four times shorter, as observed in the Pickering series. Fowler objected, however, that
the ratio of the Rydberg constants for singly-ionised helium and hydrogen was not 4,
but 4.00163.
Bohr realised that the problem arose from neglecting the contribution of the mass of the
nucleus to the computation of the moments of inertia of the hydrogen atom and the
helium ion.
The Rydberg Constants for H and He+
If the angular velocity of the electron and the nucleus about their centre of mass is ω,
the condition for the quantisation of angular momentum is
nh
= µωR2
2π
where µ = memN /(me + mN ) is the reduced mass of the atom, or ion, which takes
account of the contributions of both the electron and the nucleus to the angular
momentum; R is their separation. Therefore, the ratio of Rydberg constants for ionised
helium and hydrogen should be
me 
RHe+
1+ M 
 = 4.00160,
= 4

m
e
RH
1+
4M
where M is the mass of the hydrogen atom. Thus, precise agreement was found
between the theoretical and laboratory estimates of the ratio of Rydberg constants for
hydrogen and ionised helium.

Einstein’s Reaction
The Bohr theory of the hydrogen atom was the first convincing application of the
quantum theory to atoms. Bohr’s results were persuasive evidence that Einstein’s
quantum theory had to be taken really seriously for processes occurring on the atomic
scale. The ‘old’ quantum theory was, however, seriously incomplete and constitutes an
uneasy mixture of classical and quantum ideas.
The results provided further strong support for Einstein’s quantum picture of elementary
processes. When Einstein heard of Bohr’s analysis of the Balmer series of hydrogen in
September 1914, Einstein remarked cautiously that Bohr’s work was very interesting,
and important if right. When Hevesy told him about the helium results, Einstein
responded,
‘This is an enormous achievement. The theory of Bohr must then be right.’
Einstein (1916) ‘On the Quantum
Theory of Radiation’
By 1916, the pendulum of scientific opinion was beginning to swing in favour of the
quantum theory, particularly following the success of the Bohr’s theory of the hydrogen
atom. These ideas fed back into Einstein’s thinking about the problems of the emission
and absorption of radiation and resulted in his famous derivation of the Planck spectrum
through the introduction of what are now called Einstein’s A and B coefficients.
Einstein showed how the Maxwell and Planck distributions can be reconciled through a
derivation of the Planck spectrum, which gives insight into what he refers to as the ‘still
unclear processes of emission and absorption of radiation by matter.’
Quantum Emission and Absorption of Radiation
The paper begins with a description of a quantum system consisting of a large number
of molecules which can occupy a discrete set of states Z1, Z2, Z3, . . . with
corresponding energies ǫ1, ǫ2, ǫ3, . . . . The relative probabilities Wn of these states
being occupied in thermodynamic equilibrium at temperature T are given by
Boltzmann’s relation
ǫn
,
Wn = gn exp −
kT
where the gn are the statistical weights, or degeneracies, of the states Zn. As Einstein
remarks forcefully in his paper
‘[this equation] expresses the farthest-reaching generalisation of Maxwell’s
velocity distribution law.’
Quantum Emission and Absorption
Consider two quantum states of the gas molecules, Zm and Zn with energies ǫm and
ǫn respectively, such that ǫm > ǫn. Following the precepts of the Bohr model, it is
assumed that a quantum of radiation is emitted if the molecule changes from the state
Zm to Zn, the energy of the quantum being hν = ǫm − ǫn. Similarly, when a photon of
energy hν is absorbed, the molecule changes from the state Zn to Zm.
The quantum description of these processes follows by analogy with the classical
processes of the emission and absorption of radiation.
Spontaneous emission Einstein notes that a classical oscillator emits radiation in the
absence of excitation by an external field. The corresponding process at the quantum
level is called spontaneous emission, and the probability of it taking place in the time
interval dt is
dW = An
m dt,
similar to the law of radioactive decay.
Induced Emission and Absorption
Induced emission and absorption By analogy with the classical case, if the oscillator is
excited by waves of the same frequency as the oscillator, it either gains or loses energy,
depending upon the phase of the wave relative to that of the oscillator, that is, the work
done on the oscillator can be either positive or negative. The magnitude of the positive
or negative work done is proportional to the energy density of the incident waves.
The quantum mechanical equivalents of these processes are those of induced
absorption, in which the molecule is excited from the state Zn to Zm, and induced
emission, in which the molecule emits a photon under the influence of the incident
radiation field. The probabilities of these processes are written:
Induced absorption dW = Bnm̺ dt,
Induced emission dW
n ̺ dt.
= Bm
Balancing Absorption and Emission
The lower indices refer to the initial state and the upper indices the final state. ̺ is the
n are constants for a particular
energy density of radiation with frequency ν. Bnm and Bm
physical processes, and are referred to as ‘changes of state by induced emission’.
We now seek the spectrum of the energy density of radiation ̺(ν) in thermal
equilibrium. The relative numbers of molecules with energies ǫm and ǫn in thermal
equilibrium are given by the Boltzmann relation and so, in order to leave the equilibrium
distribution unchanged under the processes of spontaneous and induced emission and
induced absorption of radiation, the probabilities must balance, that is,
n
gne−ǫn/kT Bnm̺ = gme−ǫm/kT (Bm
̺ + An
m)} .
|
|
{z
}
{z
absorption
emission
Derivation of Planck’s Radiation Law
In the limit T → ∞, the radiation energy density ̺ → ∞, and the induced processes
dominate the equilibrium. Therefore, allowing T → ∞ and An
m = 0,
n
gnBnm = gmBm
.
The equilibrium spectrum ̺ can therefore be written
n
An
/Bm
m
.
̺=
ǫm − ǫn
−1
exp
kT
But, this is Planck’s radiation law. Suppose we consider only Wien’s law, which is
known to be the correct expression in the frequency range in which light can be
considered to consist of photons. Then, in the limit ǫm − ǫn/kT ≫ 1,
ǫm − ǫn
An
exp
−
̺= m
n
Bm
kT
hν
∝ ν 3 exp −
kT
.
The Values of Einstein’s A and B Coefficients
Therefore, we find the following ‘thermodynamic’ relations
An
m
3,
∝
ν
n
Bm
ǫm − ǫn = hν.
The value of the constant can be found from the Rayleigh-Jeans limit of the black-body
spectrum, ǫm − ǫn/kT ≪ 1.
An
8πν 2
m kT
kT
=
̺(ν) =
n hν
c3
Bm
and so
An
8πhν 3
m
.
=
n
3
Bm
c
The importance of these relations between the A and B coefficients is that they are
n
m
associated with atomic processes at the microscopic level. Once An
m or Bm or Bn is
known, the other coefficients can be found immediately.
Einstein’s Real Motivation
Einstein now used these results to determine how the motions of molecules would be
affected by the emission and absorption of quanta. The analysis was similar to his
earlier studies of Brownian motion, but now applied to the case of quanta interacting
with molecules. The quantum nature of the processes of emission and absorption were
essential features of his argument.
He found the key result that, when a molecule emits or absorbs a quantum hν, there
must be a positive or negative change in the momentum of the molecule of magnitude
|hν/c|, even in the case of spontaneous emission. In Einstein’s words,
‘There is no radiation of spherical waves. In the spontaneous emission
process, the molecule suffers a recoil of magnitude hν/c in a direction that, in
the present state of the theory, is determined only by ‘chance’.’
Millikan (1916)
“We are confronted, however, by the
astonishing situation that these facts
were correctly and exactly predicted
nine years ago by a form of quantum
theory which has now been generally
abandoned.”
In 1916, Millikan published his results of
measurements of the dependence of the
photoelectric effect upon the frequency of
the incident radiation.
He refers to Einstein’s ‘bold, not to say
reckless, hypothesis of an
electromagnetic light corpuscle of
energy hν, which flies in the face of
the thoroughly established facts of
interference.’
The End of the Story
• In 1923, Arthur Holly Compton discovered the Compton effect, the scattering of
X-rays by electrons, which could only be explained by assuming that the X-rays are
quanta, or photons, with energy E = hν and momentum p = hν/c.
• In 1924-6, Erwin Schödinger and Werner Heisenberg discover wave and matrix
mechanics respectively, which soon became recognised as the foundations of
quantum mechanics.
Einstein’s Achievement