{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

2004AugQualSTAT - Ben Sauerwine Practice for Qualifying...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon