2004AugQualSTAT - Ben Sauerwine Practice for Qualifying...

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Unformatted text preview: Ben Sauerwine Practice for Qualifying Exams Problem Source: CMU Qualifying Exam August 2004 (1) (a) Write down the canonical partition function for a quantum system for the energy of state s, and q Z s E T k B 1 = β ( is Boltzmann’s constant, T is the absolute temperature). Next write down the classical partition function appropriate for a monatomic gas of N identical non-relativistic particles of mass m in a volume V as an integral involving the Hamiltonian H. B k c Z ∑ − = s E q s e Z β where the N and V dependencies appear in the energy states and analogously, ( ) ∫ ∫ − = q pd d e h N Z N N q p H N c 3 3 , 3 ! 1 v v β (b) Beginning with appropriate expressions for the averages E and 2 E , show that the variance of the energy E can be written in the form ( ) N V E E E , 2 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ − = − β for both the quantum and classical systems considered in part (a). First, note that ( ) 2 2 2 2 2 2 E E E E E E E E − = + − = − . Further, ∑ ∑ ∑ ∑ − − − − = = s E s E s q s E s E s q s s s s e e E E e e E E β β β β 2 2 and, ( ) ( ) ( ) ( ) ∫∫ ∫∫ − − = = q pd d e q p H h N E q pd d e q p H h N E N N q p H N c N N q p H N c 3 3 , 2 3 2 3 3 , 3 , ! 1 , ! 1 v v v v v v v v β β Now: 2 2 , 2 2 2 2 , E E E so E E e E e e E e e E E N V q s E s s E s E s s E s E s N V q s s s s s − = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ∂ ∂ − + − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + − = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ∂ ∂ ∑ ∑ ∑ ∑ ∑ − − − − −...
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This note was uploaded on 04/07/2012 for the course PHYSICS 767 taught by Professor Dr.jaouni during the Spring '12 term at Abu Dhabi University.

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2004AugQualSTAT - Ben Sauerwine Practice for Qualifying...

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