HW1 - Physics 127B – Homework Set 1(due January 20...

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Unformatted text preview: Physics 127B – Homework Set 1: (due January 20) Problem 1: Brute force cummulant expansion for expectation val- ues Consider a classical statmech system with a Hamiltonian (energy) H = H + U , (1) where we will treat H as an “unperturbed” Hamiltonian and U as a small “perturbation” (more precisely, we will treat βU as small). Consider calculating expectation value of an observable A ( A ) = Tr[ e- βH A ] Tr[ e- βH ] = ( e- βU A ) ( e- βU ) , (2) where ( X ) ≡ Tr[ e- βH X ] Tr[ e- βH ] denotes expectation value of X with respect to the unperturbed problem; “Tr” denotes integration or summation over appropriate degrees of freedom, e.g., integration over { p i ,q i } for gasses or summation over spin states for magnetic systems, but need to be specified in detail here. By expanding the numerator and denominator in powers of βU , generate series for the expectation value of A up to second order in β : ( A ) = ( A ) − β ( A ∗ U ) c, + β 2 2 ( A ∗ U 2 ) c, + ... , (3) finding expressions for the first two so-called joint cummulants ( A ∗ U ) c, and ( A ∗ U 2 ) c, . Problem 2: High temperature expansions for the Ising model Consider an interacting Ising model of spins S i = ± 1 on a d-dimensional simple cubic lattice in which each spin interacts with strength J with its nearest neighbors, so that the Hamiltonian is H = 1 2 J N summationdisplay i =1 2 d summationdisplay...
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This note was uploaded on 03/07/2012 for the course PHYS 127b taught by Professor Olexeimotrunich during the Winter '11 term at Caltech.

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HW1 - Physics 127B – Homework Set 1(due January 20...

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