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INDEPENDENT and NON-INTERACTING harmonic oscillators

Calculate the canonical ensemble partition function for a collection of INDEPENDENT and NON-INTERACTING harmonic oscillators. Assume the energy spacing between adjacent states in an oscillator is LARGE compared to kT. Calculate in detail; do not skip steps and state all assumptions/approximations.

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The energy levels of a harmonic oscillator are given by:

E_k = (k + 1/2)h-bar omega. (1)

where k>= 0 is an integer. In the following I will denote the state of an oscillator by this integer.

Consider the system of N oscillators. To specify the states of this system you have to specify in which state each of the oscillators is. So, you need N integers to do that.

If oscillator number 1 is in state k1, oscillator number 2 is in state k2, ..., oscillator number p is in state kp then the total energy of this state is:

E_{k1,k2,...kN} = E_k1 + E_k2 + ... E_kN (2)

The partition function of a system of N noninteracting oscillators is thus given by:

Z_N = Sum_{k1, k2, k3,....,kN} Exp[-beta E_{k1,k2,k3...,k_N}] (3)

where the sum over the ki ranges from zero to infinity
and beta = 1/(kT)

First we simply (3). Substituting (2) in the exponent in (3) gives:

Exp[-beta E_{k1,k2,k3...,k_N}] = Exp[-beta(E_k1 + E_k2 + ...+E_kN)]=
Exp[-beta E_k1] x Exp[-beta ...

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A detailed solution is given.