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Unformatted text preview: Physics 212 Problem Set One 1. Perturbed Square Well Consider a one-dimensional system consisting of a particle of mass m and charge e in the potential well V ( x ) shown (see figure). 6 6 - a 2- a 4 a 4 a 2- x 6 ? b = 2 h 2 8 ma 2 V ( x ) The potential energy function is V ( x ) = , x <- a 2 ,x > a 2 ,- a 2 < x <- a 4 , a 4 < x < a 2 b,- a 4 < x < a 4 . Suppose that b = 2 h 2 8 ma 2 and is sufficiently small that we can treat this problem by means of perturbation theory. (a) Write expressions for the unperturbed ( b =0) eigenfunctions and energy eigenvalues. (b) Obtain an expression for the first-order perturbed energy of the ground state of the system, expressing your result in terms of b and fundamental constants. (c) Write an expression for the first-order perturbed wave function of the ground state in terms of the complete set of unperturbed functions of part (a). (You need not evaluate matrix elements in your expression.) (d) What is the time-dependent stationary-state wave function for the ground state of the unperturbed system?(d) What is the time-dependent stationary-state wave function for the ground state of the unperturbed system?...
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