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

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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 0 + U , (1) where we will treat H 0 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 ) 0 ( e - βU ) 0 , (2) where ( X ) 0 Tr[ e - βH 0 X ] Tr[ e - βH 0 ] 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 ) 0 β ( A U ) c, 0 + β 2 2 ( A U 2 ) c, 0 + . . . , (3) finding expressions for the first two so-called joint cummulants ( A U ) c, 0 and ( A U 2 ) c, 0 . 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 δ =1 S i S i + δ , (4) where the sum over δ runs over the 2 d nearest neighbors, and i + δ denotes these nearest neighbors.
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