Editing Adiabatic invariant (section)

== Classical mechanics – action variables ==
[[File:Adiabatic-pendulum.png |thumb|200px|right|alt=Forced pendulum|Pendulum with extra small vibration, where <math>\omega(t) = \sqrt{g/L(t)} \approx \sqrt{g/L_0},</math> and <math>L(t) \approx L_0 + \varepsilon(t).</math>]]
Suppose that a Hamiltonian is slowly time-varying, for example, a one-dimensional harmonic oscillator with a changing frequency:
<math display="block">
H_t(p, x) = \frac{p^2}{2m} + \frac{m \omega(t)^2 x^2}{2}.
</math>

The [[action-angle variables|action]] ''J'' of a classical orbit is the area enclosed by the orbit in phase space:
<math display="block">
J = \int_0^T p(t) \,\frac{dx}{dt} \,dt.
</math>

Since ''J'' is an integral over a full period, it is only a function of the energy. When the Hamiltonian is constant in time, and ''J'' is constant in time, the canonically conjugate variable <math>\theta</math> increases in time at a steady rate:
<math display="block">
\frac{d\theta}{dt} = \frac{\partial H}{\partial J} = H'(J).
</math>

So the constant <math>H'</math> can be used to change time derivatives along the orbit to partial derivatives with respect to <math>\theta</math> at constant ''J''. Differentiating the integral for ''J'' with respect to ''J'' gives an identity that fixes <math>H'</math>:
<math display="block">
\frac{dJ}{dJ } = 1 = \int_0^T \left(\frac{\partial p}{\partial J} \frac{dx}{dt} +
 p \frac{\partial}{\partial J} \frac{dx}{dt}\right) \,dt =
H' \int_0^T \left(\frac{\partial p}{\partial J} \frac{\partial x}{\partial \theta} - \frac{\partial p}{\partial \theta} \frac{\partial x}{\partial J}\right) \,dt.
</math>

The integrand is the [[Poisson bracket]] of ''x'' and ''p''. The Poisson bracket of two canonically conjugate quantities, like ''x'' and ''p'', is equal to 1 in any canonical coordinate system. So
<math display="block">
1 = H' \int_0^T \{x, p\} \,dt = H' T,
</math>
and <math>H'</math> is the inverse period. The variable <math>\theta</math> increases by an equal amount in each period for all values of ''J''{{snd}} it is an angle variable.

=== Adiabatic invariance of ''J'' ===

The Hamiltonian is a function of ''J'' only, and in the simple case of the harmonic oscillator,
<math display="block">
H = \omega J.
</math>

When ''H'' has no time dependence, ''J'' is constant. When ''H'' is slowly time-varying, the rate of change of ''J'' can be computed by re-expressing the integral for ''J'':
<math display="block">
J = \int_0^{2\pi} p \frac{\partial x}{\partial \theta} \,d\theta.
</math>

The time derivative of this quantity is
<math display="block">
\frac{dJ}{dt} = \int_0^{2\pi} \left(\frac{dp}{dt} \frac{\partial x}{\partial \theta} +
 p \frac{d}{dt} \frac{\partial x}{\partial \theta}\right) \,d\theta.</math>

Replacing time derivatives with theta derivatives, using <math>d\theta = \omega \, dt,</math> and setting <math>\omega := 1</math> without loss of generality (<math>\omega</math> being a global multiplicative constant in the resulting time derivative of the action) yields 
<math display="block">
\frac{dJ}{dt} = \int_0^{2\pi} \left(\frac{\partial p}{\partial \theta} \frac{\partial x}{\partial \theta} +
 p \frac{\partial}{\partial \theta} \frac{\partial x}{\partial \theta}\right) \,d\theta.
</math>

So as long as the coordinates ''J'', <math>\theta</math> do not change appreciably over one period, this expression can be integrated by parts to give zero. This means that for slow variations, there is no lowest-order change in the area enclosed by the orbit. This is the adiabatic invariance theorem{{snd}} the action variables are adiabatic invariants.

For a harmonic oscillator, the area in phase space of an orbit at energy ''E'' is the area of the ellipse of constant energy,
<math display="block">
E = \frac{p^2}{2m} + \frac{m\omega^2 x^2}{2}.
</math>

The ''x'' radius of this ellipse is <math>\sqrt{2E/\omega^2m},</math> while the ''p'' radius of the ellipse is <math>\sqrt{2mE}</math>. Multiplying, the area is <math>2\pi E/\omega</math>. So if a pendulum is slowly drawn in, such that the frequency changes, the energy changes by a proportional amount.

=== Old quantum theory ===

After Planck identified that Wien's law can be extended to all frequencies, even very low ones, by interpolating with the classical equipartition law for radiation, physicists wanted to understand the quantum behavior of other systems.

The Planck radiation law quantized the motion of the field oscillators in units of energy proportional to the frequency:
<math display="block">
E = h f = \hbar \omega.
</math>

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 [[Einstein solid|quantized oscillators]]. This model explained why the specific heat of solids approached zero at low temperatures, instead of staying fixed at <math>3k_\text{B},</math> as predicted by classical [[equipartition theorem|equipartition]].

At the [[Solvay conference]], the question of quantizing other motions was raised, and [[Hendrik Lorentz|Lorentz]] pointed out a problem, known as [[Rayleigh–Lorentz pendulum]]. If you consider a quantum pendulum whose string is shortened very slowly, the quantum number of the pendulum cannot change because at no point is there a high enough frequency to cause a transition between the states. But the frequency of the pendulum changes when the string is shorter, so the quantum states change energy.

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. The conclusion was that the quantities to quantize must be adiabatic invariants.

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. Since the action variable in the harmonic oscillator is an integer, the general condition is
<math display="block">
\int p \, dq = n h.
</math>

This condition was the foundation of the [[old quantum theory]], which was able to predict the qualitative behavior of atomic systems. The theory is inexact for small quantum numbers, since it mixes classical and quantum concepts. But it was a useful half-way step to the [[matrix mechanics|new quantum theory]].