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The Hamiltonian of the one dimensional harmonic oscillator is:
H = p^2/(2m) + 1/2 m omega^2 x^2
The partition function in the classical regime can be computed as follows. We use that the number of quantum states in a range dp of momentum space and dx in configuration space is dpdx/h. The partition function for a single oscillator can then be written as:
Z1 = Integral over space and momentum of dpdx/h exp[-beta(p^2/(2m) + 1/2 m omega^2 x^2)] =
Integral over p from minus infinity to plus infinity of dp/h exp[-beta p^2/(2m)] times
Integral over x from minus to plus infinity of dx exp(-beta 1/2 m omega^2 x^2)
We can evaluate these integrals by using:
Integral from minus to plus infinity of exp(-s t^2)dt = sqrt(pi/s)
We then get:
Integral over p from minus infinity to plus infinity of dp/h exp[-beta p^2/(2m)] =
1/h sqrt[2 m pi/beta]
Integral over x from minus to plus infinity of dx exp(-beta 1/2 m omega^2 x^2) =
1/omega sqrt[2 pi/(m beta)]
The partition function is thus given by:
Z1 = 1/(h omega) sqrt[2 m pi/beta] sqrt[2 pi/(m beta)] = 2 pi/(h omega beta) = 1/(beta hbar omega) =
k T/(hbar omega)
The partition function for N distinguishable oscillators is:
Z = Z1^N
The free energy is:
F = - k T ...
The problem is worked out in detail from first principles.
Derive the general equations for the internal energy, entropy, and Helmholtz free energy for a general system in terms of the partition function. Derive the free energy and heat capacity for the Einstein model for a crystal. Derive the high and low temperature behavior of the heat capacity.
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