# p4-342 - ~ N satisﬁes the following commutation relations...

This preview shows page 1. Sign up to view the full content.

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
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ~ N satisﬁes the following commutation relations with the angular momentum generators d ~ N dt = h H, ~ N i = 0 , ± L i ,N j ² = i ~ e ijk N k , ± N i ,N j ² = i ~ e ijk (-2 mH ) L k where ~ N = 1 2 ³ ~ 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...
View Full Document

## This note was uploaded on 02/27/2010 for the course PHYSICS 342 taught by Professor Sghy during the Spring '10 term at King's College London.

Ask a homework question - tutors are online