composed of packets that change energy and frequency in the same way. The entropy of a Wien gas scales as the volume to the power N, where N is the number of packets. 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. Classical mechanics – action variables Suppose that a Hamiltonian is slowly time-varying, for example, a one-dimensional harmonic oscillator with a changing frequency: The action J of a classical orbit is the area enclosed by the orbit in phase space: 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 increases in time at a steady rate: So the constant can be used to change time derivatives along the orbit to partial derivatives with respect to at constant J. Differentiating the integral for J with respect to J gives an identity that fixes : 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 and is the inverse period. The variable increases by an equal amount in each period for all values of J 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, 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: The time derivative of this quantity is Replacing time derivatives with theta derivatives, using and setting without loss of generality ( being a global multiplicative constant in the resulting time derivative of the action) yields So as long as the coordinates J, 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 the action variables are adiabatic invariants. For a harmonic oscillator, the area in phase space of an