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= Old quantum theory =
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Introduction
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The old quantum theory is a collection of results from the years
1900-1925 which predate modern quantum mechanics. The theory was never
complete or self-consistent, but was rather a set of heuristic
corrections to classical mechanics. The theory is now understood as
the semi-classical approximation to modern quantum mechanics.
The main tool of the old quantum theory was the Bohr-Sommerfeld
quantization condition, a procedure for selecting out certain states
of a classical system as allowed states: the system can then only
exist in one of the allowed states and not in any other state.
History
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The old quantum theory was instigated by the 1900 work of Max Planck
on the emission and absorption of light, and began in earnest after
the work of Albert Einstein on the specific heats of solids. Einstein,
followed by Debye, applied quantum principles to the motion of atoms,
explaining the specific heat anomaly.
In 1913, Niels Bohr identified the correspondence principle and used
it to formulate a model of the hydrogen atom which explained the line
spectrum. In the next few years Arnold Sommerfeld extended the quantum
rule to arbitrary integrable systems making use of the principle of
adiabatic invariance of the quantum numbers introduced by Lorentz and
Einstein. Sommerfeld made a crucial contribution by quantizing the
z-component of the angular momentum, which in the old quantum era was
called 'space quantization' (Richtungsquantelung). This allowed the
orbits of the electron to be ellipses instead of circles, and
introduced the concept of quantum degeneracy. The theory would have
correctly explained the Zeeman effect, except for the issue of
electron spin.
Sommerfeld's model was much closer to the modern quantum mechanical
picture than Bohr's.
Throughout the 1910s and well into the 1920s, many problems were
attacked using the old quantum theory with mixed results. Molecular
rotation and vibration spectra were understood and the electron's spin
was discovered, leading to the confusion of half-integer quantum
numbers. Max Planck introduced the zero point energy and Arnold
Sommerfeld semiclassically quantized the relativistic hydrogen atom.
Hendrik Kramers explained the Stark effect. Bose and Einstein gave the
correct quantum statistics for photons.
Kramers gave a prescription for calculating transition probabilities
between quantum states in terms of Fourier components of the motion,
ideas which were extended in collaboration with Werner Heisenberg to a
semiclassical matrix-like description of atomic transition
probabilities. Heisenberg went on to reformulate all of quantum theory
in terms of a version of these transition matrices, creating matrix
mechanics.
In 1924, Louis de Broglie introduced the wave theory of matter, which
was extended to a semiclassical equation for matter waves by Albert
Einstein a short time later. In 1926 Erwin Schrödinger found a
completely quantum mechanical wave-equation, which reproduced all the
successes of the old quantum theory without ambiguities and
inconsistencies. Schrödinger's wave mechanics developed separately
from matrix mechanics until Schrödinger and others proved that the two
methods predicted the same experimental consequences. Paul Dirac later
proved in 1926 that both methods can be obtained from a more general
method called transformation theory.
In the 1950s Joseph Keller updated Bohr-Sommerfeld quantization using
Einstein's interpretation of 1917,The Collected Papers of Albert
Einstein, vol. 6, A. Engel, trans., Princeton U. Press,
Princeton, NJ (1997), p. 434 now known as Einstein-Brillouin-Keller
method. In 1971, Martin Gutzwiller took into account that this method
only works for integrable systems and derived a semiclassical way of
quantizing chaotic systems from path integrals.
Basic principles
======================================================================
The basic idea of the old quantum theory is that the motion in an
atomic system is quantized, or discrete. The system obeys classical
mechanics except that not every motion is allowed, only those motions
which obey the 'quantization condition':
:
\oint\limits_{H(p,q)=E} p_i \, dq_i = n_i h
where the p_i are the momenta of the system and the q_i are the
corresponding coordinates. The quantum numbers n_i are 'integers' and
the integral is taken over one period of the motion at constant energy
(as described by the Hamiltonian). The integral is an area in phase
space, which is a quantity called the action and is quantized in units
of Planck's (unreduced) constant. For this reason, Planck's constant
was often called the 'quantum of action'.
In order for the old quantum condition to make sense, the classical
motion must be separable, meaning that there are separate coordinates
q_i in terms of which the motion is periodic. The periods of the
different motions do not have to be the same, they can even be
incommensurate, but there must be a set of coordinates where the
motion decomposes in a multi-periodic way.
The motivation for the old quantum condition was the correspondence
principle, complemented by the physical observation that the
quantities which are quantized must be adiabatic invariants. Given
Planck's quantization rule for the harmonic oscillator, either
condition determines the correct classical quantity to quantize in a
general system up to an additive constant.
This quantization condition is often known as the 'Wilson-Sommerfeld
rule', proposed independently by William Wilson and Arnold Sommerfeld.
Thermal properties of the harmonic oscillator
===============================================
The simplest system in the old quantum theory is the harmonic
oscillator, whose Hamiltonian is:
:
H= {p^2 \over 2m} + {m\omega^2 q^2\over 2}.
The old quantum theory yields a recipe for the quantization of the
energy levels of the harmonic oscillator, which, when combined with
the Boltzmann probability distribution of thermodynamics, yields the
correct expression for the stored energy and specific heat of a
quantum oscillator both at low and at ordinary temperatures. Applied
as a model for the specific heat of solids, this resolved a
discrepancy in pre-quantum thermodynamics that had troubled
19th-century scientists. Let us now describe this.
The level sets of 'H' are the orbits, and the quantum condition is
that the area enclosed by an orbit in phase space is an integer. It
follows that the energy is quantized according to the Planck rule:
:
E= n\hbar \omega,
\,
a result which was known well before, and used to formulate the old
quantum condition. This result differs by \frac{1}{2}\hbar \omega from
the results found with the help of quantum mechanics. This constant is
neglected in the derivation of the 'old quantum theory', and its value
cannot be determined using it.
The thermal properties of a quantized oscillator may be found by
averaging the energy in each of the discrete states assuming that they
are occupied with a Boltzmann weight:
:
U = {\sum_n \hbar\omega n e^{-\beta n\hbar\omega} \over \sum_n
e^{-\beta n \hbar\omega}} = {\hbar \omega e^{-\beta\hbar\omega} \over
1 - e^{-\beta\hbar\omega}},\;\;\;{\rm where}\;\;\beta = \frac{1}{kT},
'kT' is Boltzmann constant times the absolute temperature, which is
the temperature as measured in more natural units of energy. The
quantity \beta is more fundamental in thermodynamics than the
temperature, because it is the thermodynamic potential associated to
the energy.
From this expression, it is easy to see that for large values of
\beta, for very low temperatures, the average energy U in the Harmonic
oscillator approaches zero very quickly, exponentially fast. The
reason is that kT is the typical energy of random motion at
temperature T, and when this is smaller than \scriptstyle \hbar\omega,
there is not enough energy to give the oscillator even one quantum of
energy. So the oscillator stays in its ground state, storing next to
no energy at all.
This means that at very cold temperatures, the change in energy with
respect to beta, or equivalently the change in energy with respect to
temperature, is also exponentially small. The change in energy with
respect to temperature is the specific heat, so the specific heat is
exponentially small at low temperatures, going to zero like
:: \exp(-\hbar\omega/kT)
At small values of \beta, at high temperatures, the average energy U
is equal to 1/\beta = kT. This reproduces the equipartition theorem of
classical thermodynamics: every harmonic oscillator at temperature 'T'
has energy 'kT' on average. This means that the specific heat of an
oscillator is constant in classical mechanics and equal to 'k'. For a
collection of atoms connected by springs, a reasonable model of a
solid, the total specific heat is equal to the total number of
oscillators times 'k'. There are overall three oscillators for each
atom, corresponding to the three possible directions of independent
oscillations in three dimensions. So the specific heat of a classical
solid is always 3k per atom, or in chemistry units, 3R per mole of
atoms.
Monatomic solids at room temperatures have approximately the same
specific heat of 3k per atom, but at low temperatures they don't. The
specific heat is smaller at colder temperatures, and it goes to zero
at absolute zero. This is true for all material systems, and this
observation is called the third law of thermodynamics. Classical
mechanics cannot explain the third law, because in classical mechanics
the specific heat is independent of the temperature.
This contradiction between classical mechanics and the specific heat
of cold materials was noted by James Clerk Maxwell in the 19th
century, and remained a deep puzzle for those who advocated an atomic
theory of matter. Einstein resolved this problem in 1906 by proposing
that atomic motion is quantized. This was the first application of
quantum theory to mechanical systems. A short while later, Peter Debye
gave a quantitative theory of solid specific heats in terms of
quantized oscillators with various frequencies (see Einstein solid and
Debye model).
=== One-dimensional potential: U=0 ===
One-dimensional problems are easy to solve. At any energy 'E', the
value of the momentum 'p' is found from the conservation equation:
:
\sqrt{2m(E - V(q))} = p
which is integrated over all values of 'q' between the classical
'turning points', the places where the momentum vanishes. The integral
is easiest for a 'particle in a box' of length 'L', where the quantum
condition is:
:
2\int_0^L p \, dq = nh
which gives the allowed momenta:
:
p= {nh \over 2L}
and the energy levels
:
E_n= {p^2 \over 2m} = {n^2 h^2 \over 8mL^2}
=== One-dimensional potential: U=Fx ===
Another easy case to solve with the old quantum theory is a linear
potential on the positive halfline, the constant confining force 'F'
binding a particle to an impenetrable wall. This case is much more
difficult in the full quantum mechanical treatment, and unlike the
other examples, the semiclassical answer here is not exact but
approximate, becoming more accurate at large quantum numbers.
:
2 \int_0^{\frac{E}{F}} \sqrt{2m(E - Fx)}\ dx= n h
so that the quantum condition is
:
{4\over 3} \sqrt{2m}{ E^{3/2}\over F } = n h
which determines the energy levels,
:
E_n = \left({3nhF\over 4\sqrt{2m}} \right)^{2/3}
In the specific case F=mg, the particle is confined by the
gravitational potential of the earth and the "wall" here is the
surface of the earth.
=== One-dimensional potential: U=½kx² ===
This case is also easy to solve, and the semiclassical answer here
agrees with the quantum one to within the ground-state energy. Its
quantization-condition integral is
:
2 \int_{-\sqrt{\frac{2E}{k}}}^{\sqrt{\frac{2E}{k}}} \sqrt{2m\left(E -
\frac12 k x^2\right)}\ dx = n h
with solution
:
E = n \frac{h}{2\pi} \sqrt{\frac{k}{m}} = n \hbar \omega
for oscillation angular frequency \omega, as before.
Rotator
=========
Another simple system is the rotator. A rotator consists of a mass 'M'
at the end of a massless rigid rod of length 'R' and in two dimensions
has the Lagrangian:
:
L = {MR^2 \over 2} \dot\theta^2
which determines that the angular momentum 'J' conjugate to \theta,
the polar angle, \scriptstyle J = MR^2 \dot\theta. The old quantum
condition requires that 'J' multiplied by the period of \theta is an
integer multiple of Planck's constant:
:
2\pi J = n h
\,
the angular momentum to be an integer multiple of \scriptstyle \hbar.
In the Bohr model, this restriction imposed on circular orbits was
enough to determine the energy levels.
In three dimensions, a rigid rotator can be described by two angles �
\scriptstyle \theta and \scriptstyle \phi, where \scriptstyle \theta
is the inclination relative to an arbitrarily chosen 'z'-axis while
\scriptstyle \phi is the rotator angle in the projection to the
'x'-'y' plane. The kinetic energy is again the only contribution to
the Lagrangian:
:
L = {MR^2\over 2} \dot\theta^2 + {MR^2\over 2}
(\sin(\theta)\dot\phi)^2
\,
And the conjugate momenta are \scriptstyle p_\theta = \dot\theta and
\scriptstyle p_\phi=\sin(\theta)^2 \dot\phi. The equation of motion
for \scriptstyle \phi is trivial: \scriptstyle p_\phi is a constant:
:
p_\phi = l_\phi
\,
which is the 'z'-component of the angular momentum. The quantum
condition demands that the integral of the constant \scriptstyle
l_\phi as \scriptstyle \phi varies from 0 to 2\pi is an integer
multiple of 'h':
:
l_\phi = m \hbar
\,
And 'm' is called the magnetic quantum number, because the 'z'
component of the angular momentum is the magnetic moment of the
rotator along the 'z' direction in the case where the particle at the
end of the rotator is charged.
Since the three-dimensional rotator is rotating about an axis, the
total angular momentum should be restricted in the same way as the
two-dimensional rotator. The two quantum conditions restrict the total
angular momentum and the 'z'-component of the angular momentum to be
the integers 'l','m'. This condition is reproduced in modern quantum
mechanics, but in the era of the old quantum theory it led to a
paradox: how can the orientation of the angular momentum relative to
the arbitrarily chosen 'z'-axis be quantized? This seems to pick out a
direction in space.
This phenomenon, the quantization of angular momentum about an axis,
was given the name 'space quantization', because it seemed
incompatible with rotational invariance. In modern quantum mechanics,
the angular momentum is quantized the same way, but the discrete
states of definite angular momentum in any one orientation are quantum
superpositions of the states in other orientations, so that the
process of quantization does not pick out a preferred axis. For this
reason, the name "space quantization" fell out of favor, and the same
phenomenon is now called the quantization of angular momentum.
Hydrogen atom
===============
The angular part of the hydrogen atom is just the rotator, and gives
the quantum numbers 'l' and 'm'. The only remaining variable is the
radial coordinate, which executes a periodic one-dimensional potential
motion, which can be solved.
For a fixed value of the total angular momentum 'L', the Hamiltonian
for a classical Kepler problem is (the unit of mass and unit of energy
redefined to absorb two constants):
:
H= { p^2 \over 2 } + {l^2 \over 2 r^2 } - {1\over r}.
Fixing the energy to be (a negative) constant and solving for the
radial momentum 'p', the quantum condition integral is:
:
2 \oint \sqrt{2E - {l^2\over r^2} + { 2\over r}}\ dr= k h
which can be solved with the method of residues, and gives a new
quantum number k which determines the energy in combination with l.
The energy is:
:
E= -{1 \over 2 (k + l)^2}
and it only depends on the sum of 'k' and 'l', which is the 'principal
quantum number' 'n'. Since 'k' is positive, the allowed values of 'l'
for any given 'n' are no bigger than 'n'. The energies reproduce those
in the Bohr model, except with the correct quantum mechanical
multiplicities, with some ambiguity at the extreme values.
The semiclassical hydrogen atom is called the Sommerfeld model, and
its orbits are ellipses of various sizes at discrete inclinations. The
Sommerfeld model predicted that the magnetic moment of an atom
measured along an axis will only take on discrete values, a result
which seems to contradict rotational invariance but which was
confirmed by the Stern-Gerlach experiment. This Bohr-Sommerfeld theory
is a significant step in the development of quantum mechanics. It also
describes the possibility of atomic energy levels being split by a
magnetic field (called the Zeeman effect).
Relativistic orbit
====================
Arnold Sommerfeld derived the relativistic solution of atomic energy
levels. We will start this derivation with the relativistic equation
for energy in the electric potential
:W={m_\mathrm{0} c^2} \left( \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} - 1
\right) - k \frac{Z e^2}{r}
After substitution u = \frac{1}{r} we get
:\frac{1}{\sqrt{1-\frac{v^2}{c^2}}} = 1 + \frac{W}{m_\mathrm{0} c^2} +
k \frac{Z e^2}{m_\mathrm{0} c^2} u
For momentum p_\mathrm{r} = m \dot{r}, p_\mathrm{\varphi} = m r^2
\dot{\varphi} and their ratio \frac{p_\mathrm{r}}{p_\mathrm{\varphi}}
= - \frac{du}{d \varphi} the equation of motion is (see Binet
equation)
:\frac{d^2 u}{d \varphi ^2} = - \left( 1 - k^2 \frac{Z^2 e^4}{c^2
p_\mathrm{\varphi}^2} \right) u + \frac{m_\mathrm{0}
kZe^2}{p_\mathrm{\varphi}^2} \left( 1+\frac{W}{m_\mathrm{0} c^2}
\right) = - \omega_\mathrm{0}^2 u + K
with solution
: u = \frac{1}{r} = K + A \cos \omega_\mathrm{0} \varphi
The angular shift of periapsis per revolution is given by
: \varphi_\mathrm{s} = 2 \pi \left(\frac{1}{\omega_\mathrm{0}} -
1\right) \approx 4 \pi^3 k^2 \frac{Z^2 e^4}{c^2 n_\mathrm{\varphi}^2
h^2}
With the quantum conditions
: \oint p_\mathrm{\varphi} \, d \varphi = 2 \pi p_\mathrm{\varphi} =
n_\mathrm{\varphi} h
and
: \oint p_\mathrm{r} \, d r = p_\mathrm{\varphi} \oint \left(
\frac{1}{r} \frac{dr}{d \varphi} \right)^2 \, d \varphi = n_\mathrm{r}
h
we will obtain energies
: \frac{W}{m_\mathrm{0} c^2} = \left( 1 + \frac{\alpha ^2
Z^2}{(n_\mathrm{r} + \sqrt{n_\mathrm{\varphi} ^2 - \alpha ^2 Z^2} )^2}
\right) ^{-1/2} - 1
where \alpha is the fine-structure constant. This solution (using
substitutions for quantum numbers) is equivalent to the solution of
the Dirac equation. Nevertheless, both solutions fail to predict the
Lamb shifts.
De Broglie waves
======================================================================
In 1905, Einstein noted that the entropy of the quantized
electromagnetic field oscillators in a box is, for short wavelength,
equal to the entropy of a gas of point particles in the same box. The
number of point particles is equal to the number of quanta. Einstein
concluded that the quanta could be treated as if they were localizable
objects
(see page 139/140), particles of light, and named them photons.
Einstein's theoretical argument was based on thermodynamics, on
counting the number of states, and so was not completely convincing.
Nevertheless, he concluded that light had attributes of both waves and
particles, more precisely that an electromagnetic standing wave with
frequency \omega with the quantized energy:
:
E = n\hbar\omega
\,
should be thought of as consisting of n photons each with an energy
\scriptstyle \hbar\omega. Einstein could not describe how the photons
were related to the wave.
The photons have momentum as well as energy, and the momentum had to
be \scriptstyle \hbar k where k is the wavenumber of the
electromagnetic wave. This is required by relativity, because the
momentum and energy form a four-vector, as do the frequency and
wave-number.
In 1924, as a PhD candidate, Louis de Broglie proposed a new
interpretation of the quantum condition. He suggested that all matter,
electrons as well as photons, are described by waves obeying the
relations.
:
p = \hbar k
or, expressed in terms of wavelength \lambda instead,
:
p = {h \over \lambda}
He then noted that the quantum condition:
:
\int p \, dx = \hbar \int k \, dx = 2\pi\hbar n
counts the change in phase for the wave as it travels along the
classical orbit, and requires that it be an integer multiple of 2\pi.
Expressed in wavelengths, the number of wavelengths along a classical
orbit must be an integer. This is the condition for constructive
interference, and it explained the reason for quantized orbits�the
matter waves make standing waves only at discrete frequencies, at
discrete energies.
For example, for a particle confined in a box, a standing wave must
fit an integer number of wavelengths between twice the distance
between the walls. The condition becomes:
:
n\lambda = 2L
\,
so that the quantized momenta are:
:
p = \frac{nh}{2L}
reproducing the old quantum energy levels.
This development was given a more mathematical form by Einstein, who
noted that the phase function for the waves: \theta(J,x) in a
mechanical system should be identified with the solution to the
Hamilton-Jacobi equation, an equation which even William Rowan
Hamilton believed to be a short-wavelength limit of a sort of wave
mechanics in the 19th century. Schrodinger then found the proper wave
equation which matched the Hamilton-Jacobi equation for the phase,
this is the famous equation.
Kramers transition matrix
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The old quantum theory was formulated only for special mechanical
systems which could be separated into action angle variables which
were periodic. It did not deal with the emission and absorption of
radiation. Nevertheless, Hendrik Kramers was able to find heuristics
for describing how emission and absorption should be calculated.
Kramers suggested that the orbits of a quantum system should be
Fourier analyzed, decomposed into harmonics at multiples of the orbit
frequency:
:
X_n(t) = \sum_{k=-\infty}^{\infty} e^{ik\omega t} X_{n;k}
The index 'n' describes the quantum numbers of the orbit, it would be
'n'-'l'-'m' in the Sommerfeld model. The frequency \omega is the
angular frequency of the orbit \scriptstyle 2\pi/T_n while 'k' is an
index for the Fourier mode. Bohr had suggested that the 'k'-th
harmonic of the classical motion correspond to the transition from
level 'n' to level 'n'�'k'.
Kramers proposed that the transition between states were analogous to
classical emission of radiation, which happens at frequencies at
multiples of the orbit frequencies. The rate of emission of radiation
is proportional to |X_k|^2, as it would be in classical mechanics.
The description was approximate, since the Fourier components did not
have frequencies that exactly match the energy spacings between
levels.
This idea led to the development of matrix mechanics.
Limitations
======================================================================
The old quantum theory had some limitations:
* The old quantum theory provides no means to calculate the
intensities of the spectral lines.
* It fails to explain the anomalous Zeeman effect (that is, where the
spin of the electron cannot be neglected).
* It cannot quantize "chaotic" systems, i.e. dynamical systems in
which trajectories are neither closed nor periodic and whose
analytical form does not exist. This presents a problem for systems as
simple as a 2-electron atom which is classically chaotic analogously
to the famous gravitational three-body problem.
However it can be used to describe atoms with more than one electron
(e.g. Helium) and the Zeeman effect.
It was later proposed that the old quantum theory is in fact the
semi-classical approximation to the canonical quantum mechanics but
its limitations are still under investigation.
Further reading
======================================================================
*
* Address to annual meeting of the Optical Society of America October
21, 1982 (Tucson AZ). Retrieved 2013-09-08.
*
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=========
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