p4-342 - ~ N satises the following commutation relations...

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Problem Set 4 Physics 342 Due February 4 Some abbreviations: S - Shankar. 1 . Let’s consider a spherical potential well. (i) Determine the energy levels for a particle with angular momentum l = 0 in the spherical potential well: U ( r ) = 0 r < a, U ( r ) = U 0 r > a. (ii) Discuss the limit in which the depth of the well U 0 goes to infinity. (iii) For the infinite potential well determine the energy levels for l > 0. You may write an implicit equation that determines the energy levels and need not solve it. 2 . Let us consider the Hamiltonian H = ~ p 2 2 m - e 2 r which describes a Coulomb interaction. (i) Show that the Runge-Lenz vector ~ N = ~ p × ~ L - me 2 r ~ r is conserved classically. Verify that ~ N · ~ L = 0. (ii) Show that in quantum mechanics, the vector
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Unformatted text preview: ~ N satises the following commutation relations with the angular momentum generators d ~ N dt = h H, ~ N i = 0 , L i ,N j = i ~ e ijk N k , N i ,N j = i ~ e ijk (-2 mH ) L k where ~ N = 1 2 ~ p ~ L-~ L ~ p -me 2 r ~ r. 3 . Two spin 1 / 2 particles interact via the Hamiltonian H = J ( S x 1 S x 2 + S y 1 S y 2 + kS z 1 S z 2 ) where J,k are constants. Find the energy levels of this system and their degeneracies. Discuss separately the cases where the spatial wavefunction is symmetric and antisymmetric under exchange of the two electrons. 4 . S15.3.3 5 . S15.3.4 6 . S15.3.5 1...
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