Home > 0877 > GR0877 Problem 100

## GR0877 Problem 100

PROBLEM STATEMENT: The partition function $Z$ in statistical mechanics can be written as

$\displaystyle Z = \sum_{r} e^{- E_r /kT}$,

where the index $r$ ranges over all possible microstates of a system and $E_r$ is the energy of microstate $r$. For a single quantum mechanical harmonic oscillator with energies

$E_{n} = \left( n + \frac{1}{2} \right)\hbar \omega$, where $n = 0, 1, 2,\dots,$

the partition function $Z$ is given by which of the following?
(A) $\displaystyle Z = e^{- \frac{1}{2}\hbar \omega /kT}$

(B) $\displaystyle Z = e^{\frac{1}{2}\hbar \omega /kT}$

(C) $\displaystyle Z = e^{\frac{1}{2}\hbar \omega /kT} - 1$

(D) $\displaystyle Z = e^{\frac{1}{2}\hbar \omega /kT} + 1$

(E) $\displaystyle Z = \frac{e^{\frac{1}{2}\hbar \omega /kT}}{e^{\hbar \omega /kT}-1}$

SOLUTION: (E) The partition function for a single QM harmonic oscillator is:
$\displaystyle Z = \sum_{n=0}^{\infty} \exp\left( -\frac{E_n}{kT}\right) = \exp\left( -\frac{\hbar \omega}{2kT}\right) \sum_{n=0}^{\infty} \exp\left( -\frac{n \hbar \omega}{kT}\right)$.
$\displaystyle \sum_{n=0}^{\infty} \exp\left( -\frac{n \hbar \omega}{kT}\right) = 1 + \exp\left( -\frac{\hbar \omega}{kT}\right) + \exp\left( -\frac{2 \hbar \omega}{kT}\right) + \dots$ which is an infinite geometric series:
$\displaystyle \sum_{n=0}^{\infty} \exp\left( -\frac{n \hbar \omega}{kT}\right) = \frac{1}{1 - \exp\left( -\frac{\hbar \omega}{kT}\right)}$. Thus, $\displaystyle Z = \frac{\exp\left( -\frac{\hbar \omega}{2kT}\right)}{1 - \exp\left( -\frac{\hbar \omega}{kT}\right)} = \frac{\exp\left( \frac{\hbar \omega}{2kT}\right)}{\exp\left( \frac{\hbar \omega}{kT}\right) -1}$.

Found a typo? Comment!

Jump to the problem

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

Want to get a full PDF with solutions? Read THIS.

The comments appending particular solutions are due to the respective users. Educational Testing Service, ETS, the ETS logo, Graduate Record Examinations, and GRE are registered trademarks of Educational Testing Service. The examination questions are © 2008 by ETS.

Categories: 0877 Tags: , , , ,
1. 09.10.2012 at 02:18

I just wanted to say thank you for your solutions. They have helped so much! I take the PGRE in 5 days, and your solutions to this exam were so, so helpful!!! I wish you all the best.

1. No trackbacks yet.