p137bp4sol - Physics 137B, Fall 2007, Moore Problem Set 4...

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Unformatted text preview: Physics 137B, Fall 2007, Moore Problem Set 4 Solutions 1. We start with Eq. (8.10) in Bransden: H (1) n + H n (0) = E (0) n (1) n + E (1) n (0) n . In Dirac ket notation this reads H | (1) n i + H | (0) n i = E (0) n | (1) n i + E (1) n | (0) n i . If we hit this equation with the bra h (1) n | we obtain h (1) n | H | (1) n i + h (1) n | H | (0) n i = E (0) n h (1) n | (1) n i + E (1) n h (1) n | (0) n i . (Note that the first order corrections | (1) n i are not normalized, so we cant simplify the first term on the right-hand side.) Rearranging, this reads h (1) n | H- E (1) n | (0) n i =-h (1) n | H- E (0) n | (1) n i . Now taking the Hermitian conjugate, and using the fact that H and H are Hermitian, we have h (0) n | H- E (1) n | (1) n i =-h (1) n | H- E (0) n | (1) n i . By Eq. (8.17), the left-hand side is precisely E (2) n . 1 2. For reference, we write the first two eigenfunctions for the one-dimensional harmonic oscillator, using Eq. (4.168) and = ( m/ h ) 1 / 2 : ( x ) = 1 / 2 e- 2 x 2 / 2 , 1 ( x ) = 2 1 / 2 2 x e- 2 x 2 / 2 . The eigenfunctions of the unperturbed two-dimensional harmonic oscillator H =- h 2 2 m 2 x 2 + 2 y 2 + 1 2 k ( x 2 + y 2 ) are just products of the one-dimensional eigenfunctions, i.e.are just products of the one-dimensional eigenfunctions, i....
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p137bp4sol - Physics 137B, Fall 2007, Moore Problem Set 4...

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