6 feb. 2017
https://en.wikipedia.org/wiki/Old_quantum_theory
old quantum theory is a collection of results from the years 1900–1925 which predate modern
quantum mechanics.
https://en.wikipedia.org/wiki/Old_quantum_theory
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 a set of
heuristic prescriptions which are now understood to be the first quantum corrections to
classical mechanics.
[1] The
Bohr model was the focus of study, and
Arnold Sommerfeld[2] 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.
The main tool was
Bohr–Sommerfeld quantization, a procedure
for selecting out certain discrete set of states of a classical
integrable motion as allowed states.
These are like the allowed orbits
of the Bohr model of the atom; the system can only be in one of these
states and not in any states in between.
- 1 Basic principles
- 2 Examples
- 3 De Broglie waves
- 4 Kramers transition matrix
- 5 Limitations of the old quantum theory
- 6 History
- 7 References
- 8 Further reading
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 old quantum condition:
where the 
are the momenta of the system and the 
are the corresponding coordinates.
The quantum numbers 
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 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
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.
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[5] 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 
with the quantized energy:
should be thought of as consisting of n photons each with an energy 
. 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 
where 
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.
or, expressed in terms of wavelength 
instead,
He then noted that the quantum condition:
counts the change in phase for the wave as it travels along the classical orbit, and requires that it be an integer multiple of 
.
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:
so that the quantized momenta are:
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: 
in a mechanical system should be identified with the solution to the Hamilton–Jacobi equation, an equation which even Hamilton considered to be the short-wavelength limit of wave mechanics.
These ideas led to the development of the Schrödinger equation.
Limitations of the old quantum theory
The old quantum theory had some limitations:[6]
- 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.
[7] It was later proposed that the old quantum theory is in fact the
semi-classical approximation to the canonical quantum mechanics
[8] but its limitations are still under investigation.
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