Adiabatic invariant

A property of a physical system, such as the entropy of a gas, that stays approximately constant when changes occur slowly is called an adiabatic invariant.

In thermodynamics, an adiabatic process is a change that occurs without heat flow; it may be slow or fast.

In quantum mechanics, an adiabatic change is one that occurs at a rate much slower than the difference in frequency between energy eigenstates.

In this case, the energy states of the system do not make transitions, so that the quantum number is an adiabatic invariant.

This determined the form of the Bohr–Sommerfeld quantization rule: the quantum number is the area in phase space of the classical orbit.

They occur slowly in comparison to the other characteristic timescales of the system of interest[1] and allow heat flow only between objects at the same temperature.

By definition, a gas is ideal when its temperature is only a function of the internal energy per particle, not the volume.

This gives a differential relationship between the changes in temperature and volume, which can be integrated to find the invariant.

The different internal motions of the gas with total energy E define a sphere, the surface of a 3N-dimensional ball with radius

Using Stirling's approximation for the gamma function, and ignoring factors that disappear in the logarithm after taking N large,

For a box of radiation, ignoring quantum mechanics, the energy of a classical field in thermal equilibrium is infinite, since equipartition demands that each field mode has an equal energy on average, and there are infinitely many modes.

This is physically ridiculous, since it means that all energy leaks into high-frequency electromagnetic waves over time.

Still, without quantum mechanics, there are some things that can be said about the equilibrium distribution from thermodynamics alone, because there is still a notion of adiabatic invariance that relates boxes of different size.

When a box is slowly expanded, the frequency of the light recoiling from the wall can be computed from the Doppler shift.

This means that the change in frequency of the light is equal to the work done on the wall by the radiation pressure.

Since moving the wall slowly should keep a thermal distribution fixed, the probability that the light has energy E at frequency f must only be a function of E/f.

This function cannot be determined from thermodynamic reasoning alone, and Wien guessed at the form that was valid at high frequency.

Wien's law implicitly assumes that light is statistically composed of packets that change energy and frequency in the same way.

This led Einstein to suggest that light is composed of localizable particles with energy proportional to the frequency.

Then the entropy of the Wien gas can be given a statistical interpretation as the number of possible positions that the photons can be in.

Suppose that a Hamiltonian is slowly time-varying, for example, a one-dimensional harmonic oscillator with a changing frequency:

The Planck radiation law quantized the motion of the field oscillators in units of energy proportional to the frequency:

The quantum can only depend on the energy/frequency by adiabatic invariance, and since the energy must be additive when putting boxes end-to-end, the levels must be equally spaced.

Einstein, followed by Debye, extended the domain of quantum mechanics by considering the sound modes in a solid as quantized oscillators.

This model explained why the specific heat of solids approached zero at low temperatures, instead of staying fixed at

At the Solvay conference, the question of quantizing other motions was raised, and Lorentz pointed out a problem, known as Rayleigh–Lorentz pendulum.

Einstein responded that for slow pulling, the frequency and energy of the pendulum both change, but the ratio stays fixed.

This is analogous to Wien's observation that under slow motion of the wall the energy to frequency ratio of reflected waves is constant.

This line of argument was extended by Sommerfeld into a general theory: the quantum number of an arbitrary mechanical system is given by the adiabatic action variable.

This condition was the foundation of the old quantum theory, which was able to predict the qualitative behavior of atomic systems.