Question

remember and use <E>=?EiPiand Pi=1/Z ?exp(-?Ei) and Z=?exp(-?Ei) with ?=1/kT.

(a) show that-dZ/d? = ?Eiexp(-?Ei) and

(b) that the average energy can be derived directly from Z like <E>=-1/ZdZ/d?

(c) show further that this simple expression is also correct <E> =-dlnZ/ d?

Answer 1

a] The partition function is given by:

expanding this gives:

differentiating this with respect to beta gives

[since ]

therefore, again writing it in summation form gives:

=>

b]

dividing both the sides by Z gives:

since Z is summed over all states, it can be taken inside the summation

but we know that the average energy is given by:

therefore,

c]

now consider the RHS term:

it is of the form:

and [using chain rule]

therefore,

.