Equipartition of Energy

Physically, if any coordinate or momentum contributes to the energy of the particle as the square of the coordinate or momentum, we associate an energy factor of ½t with each coordinate or momentum. This is known as equipartition of energy. To see this, look at the phase space of the particle. Start with one dimensional motion. Then the phase space is

drawing

This can be used to show the total motion of the particle. Notice that the phase space is two dimensional. If the particle is free to move in 3 dimensions, we see that the associated phase space has 6 dimensions. Thus, for N particles, the phase space has 6N dimensions. In general, for f degrees of freedom we need coordinates q₁, q₂, ..., q_f and their corresponding momenta p_l, p₂, ..., p_f to specify the state. However, quantum mechanically we know that it is impossible to totally specify the position and momentum of the state, so we end up with a box of volume h in the phase space for a specific state. If the volume of the total accessible phase space is W, the multiplicity of the space is

g = W/h

and the entropy becomes

s = ln(W) - ln(h)

Recall that the Hamiltonian for the system is

H = T + V

and this is the total energy of the system. Then the probability is

(8.1)

We can use this to calculate the expectation value of our measurable variables as normal.

Example:
Consider a one dimensional harmonic oscillator. The energy function is given by

formula

So the average energy is

formula

Both of these terms are integrals of the form

formula

Evaluating these integrals, we get

and

thus,

formula

Making the identification a = 1/t, we see that each integral contributes a factor of ½t for a total energy of t, which we expected from the equipartition theorem.