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

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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 U as small). Consider calculating expectation value of an observable A ( A ) = Tr[ e- H A ] Tr[ e- H ] = ( e- U A ) ( e- U ) , (2) where ( X ) Tr[ e- H X ] Tr[ e- H ] 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 U , generate series for the expectation value of A up to second order in : ( A ) = ( A ) ( A U ) c, + 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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HW1 - Physics 127B Homework Set 1: (due January 20) Problem...

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