Quick partition function question. (Stat. Mech.)

[camzie69]

Physical interpretation of the partition function. Consider a single-particle quantum system whose states are labeled with an index i = 1, 2, 3, ... with corresponding energies E1, E2, E3, ... . Set the zero of energy at the ground state energy so that E1 = 0. Argue that, if the absolute temperature T is such that kT = En, where n is some integer, then n is the approximate value of the partition function (15.4). Therefore, the partition function is equal to the number of states which are likely to be occupied.

15.4 $$\sum$$ e^-Ej/kT

The Attempt at a Solution

I tried expanding terms and plugging in for kT=En yielding this but not sure where or if to proceed from here...its not correct right now...

e^-E1/En + e^-E2/En ... + e^-En/En

This gives me 1 for the first term which is a promising first step because if all the exponentials equal one I have the correct solution 1+1+1+...=n but I'm not sure if this is the case of much less how to prove this is the case.

Should this be posted in the mathematics section?

 

[BigJDubs]

The partition function is the sum of all the possible states of a system weighted by its respective energy. In the case of a single-particle system, this means that every state i has an energy Ei associated with it, and we can use the partition function to calculate the probability of any given state being occupied at a certain temperature T. When kT = En, the partition function becomes the sum of all the exponentials of the energy divided by the temperature, which is equal to the number of states that are likely to be occupied (n). The physical interpretation of the partition function is that it is equal to the number of states that are likely to be occupied at a particular temperature.