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Unformatted text preview: PHYS852 Quantum Mechanics II, Spring 2010 HOMEWORK ASSIGNMENT 5: Solutions Topics covered: rotation with spin, exchange symmmetry 1. The Hamiltonian for the deuteron, a bound-state of a proton and neutron, may be written in the form H = P 2 p 2 M p + P 2 n 2 M n + V 1 ( R ) + V 2 ( R ) ~ S p · ~ S n , (1) where R is the relative radial coordinate. Both are spin-1/2 particles, but they are not identical. (a) The total angular momentum operator is ~ S = ~ S p + ~ S n . The state | s p s n sm i is the simultaneous eigenstate of ~ S p , ~ S n , S 2 , and S z . What are the allowed values of the total spin quantum number s ? For each s-value, what are the allowed m quantum numbers. With s p = 1 / 2 and s n = 1 / 2, we find s min = | s p- s n | = 0, and s max = s p + s n = 1, so the allowed values of s are 0 , 1. For s = 0, only m = 0 is allowed, while for s = 1, we can have m =- 1 , , 1. (b) Show that | s p s n sm i is an eigenstate of ~ S p · ~ S n , and give the corresponding eigenvalue. Hint, use the fact that S 2 = ( ~ S p + ~ S n ) · ( ~ S p + ~ S n ). We have S 2 = S 2 p + 2 ~ S p · ~ S n + S 2 n (2) solving for ~ S p · ~ S n gives ~ S p · ~ S n = 1 2 ( S 2- S 2 p- S 2 n ) (3) As | s p s n sm i an eigenstate of S 2 , S 2 p and S 2 n , then it must also be an eigenstate of ~ S p · ~ S n with eigenvalue ~ 2 2 ( s ( s + 1)- 3). (c) Give ten distinct quantum numbers that can be assigned to an eigenstates of this H . Note that this includes s p and s n , even though they can never change....
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