p137bp2sol - Physics 137B Fall 2007 Moore Problem Set 2...

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Unformatted text preview: Physics 137B, Fall 2007, Moore Problem Set 2 Solutions 1. (a) With V ( r ) = Aδ ( r ), the first-order energy shift to the unperturbed energy eigenstate | nlm is E (1) nlm = nlm | V | nlm = d 3 r ψ * nlm ( r ) Aδ ( r ) ψ nlm ( r ) = A | ψ nlm (0) | 2 . For l = 0, using the hint that ψ n 00 (0) = 2 √ 4 π ( na )- 3 / 2 , this gives E (1) n 00 = A πa 3 n 3 . Remark: The 3-dimensional delta function δ ( r ) is not the same as a delta function in the radial coordinate r , which would be denoted δ ( r ). The 3-dimensional delta function δ ( r ) is something that’s meant to be integrated over 3-dimensional regions, i.e. d 3 r δ ( r ) = 1. That’s why it’s sometimes denoted δ 3 ( r ). (b) For this problem let | nlm denote the unperturbed energy eigenstates, and | φ nlm the perturbed eigenstates. The first order correction to the state | nlm is given by the usual formula | φ nlm = | nlm + n l m = nlm | n l m n l m | V | nlm E n- E n . Notice that the sum is over all possible states | n l m . However, the matrix element n l m | V | nlm = d 3 r ψ * n l m ( r ) Aδ ( r ) ψ nlm ( r ) = Aψ * n l m (0) ψ nlm (0) 1 vanishes unless l = m = 0 and l = m = 0. To see this, remember that the wavefunction factorizes into a radial part and a spherical harmonic, ψ nlm ( r ) = R nl ( r ) Y lm ( θ, φ ). At r = 0, the wavefunction should be independent of the angles θ, φ , since r = 0 is rotationally symmetric. That means that either Y lm ( θ, φ ) is constant or R nl (0) = 0. The only constant spherical harmonic is(0) = 0....
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p137bp2sol - Physics 137B Fall 2007 Moore Problem Set 2...

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