Question
Asked Mar 2, 2020
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It's a statistical mechanics question.

In lecture we derived the canonical partition function QN(V,N) for the idea gas using a factorization method. One can also directly compute it
by taking the Laplace transform of the microcanonical partition function Q(E,V,N). Using our result from lecture,
Q(E,V,N) = [3 (2TME)3/2 ]N
[(3N/2) - 1]! N!
(Δ/Ε)
compute directly the Laplace transform,
QN(T,V) = |
dE
A Q(E,V,N)e-BE
and show that you get the same result for QN(T,V) as found in lecture using the factorization method.
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In lecture we derived the canonical partition function QN(V,N) for the idea gas using a factorization method. One can also directly compute it by taking the Laplace transform of the microcanonical partition function Q(E,V,N). Using our result from lecture, Q(E,V,N) = [3 (2TME)3/2 ]N [(3N/2) - 1]! N! (Δ/Ε) compute directly the Laplace transform, QN(T,V) = | dE A Q(E,V,N)e-BE and show that you get the same result for QN(T,V) as found in lecture using the factorization method.

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Expert Answer

Step 1

Write the microcanonical partition function (E,V,N) and take the Laplace transform of it.

Advanced Physics homework question answer, step 1, image 1

Substitute the expression given for (E,V,N) in the above equation.

Advanced Physics homework question answer, step 1, image 2

Step 2

Use the gamma integral and gamma function property to calculate the above integral, i.e.,

Advanced Physics homework question answer, step 2, image 1

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