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The basic equations:
Pi = (1/Z)exp(-βEi)
Z = Σie-βEi
E = -∂lnZ/∂β
KBT = 1/β
S = βE + lnZ
A = -KBTlnZ
The Harmonic Oscillator
-----------------------
Consider the classical harmonic oscillator.
E = p2/2m + kx2/2 (KE + PE)
Z = ∫dxdpexp(-β[p2/2m + kx2/2]
= ∫dpexp(-βp2/2m) ∫exp(-βkx2/2)dx
= √(2πm/β)√(2π/βk)
= 2πm/βω where ω = √(k/m)
Now,
-lnZ = -ln(2π/ω) + lnβ
and.
E = -∂lnZ/∂β
= 1/β
= KBT
If we only consider the kinetic energy term then,
-lnZ = -(1/2)ln(2πm) + (1/2)lnβ
and.
E = -∂lnZ/∂β
= (1/2)β
= KBT/2
Likewise, if we only consider the potential energy term
then,
-lnZ = -(1/2)ln(2π/k) + (1/2)lnβ
and.
E = -∂lnZ/∂β
= (1/2)β
= KBT/2
This is in agreement with the EQUIPARTION OF ENERGY theorem
that states in a system in thermal equilibrium there will be an
average energy of KBT/2 associated with each degree of freedom.
Thus, for a system of n kinetic degrees of freedom and m
associated potential energy degrees of freedom, the total
average energy of the system will be (1/2)(n + m)KBT.
Now consider the quantum harmonic oscillator.
E = nhω
Z = Σnexp(-βnhω)
= 1 + exp(-βhω) +(exp(-βhω))2 + ...
This is the geometric series:
= 1/(1 - exp(-βhω))
Now,
-lnZ = ln(1 - exp(-βhω))
and,
E = -∂lnZ/∂β
= hω/(exp(βhω) - 1)