PHYS852 Quantum Mechanics II, Spring 2010HOMEWORK ASSIGNMENT 5Topics covered: rotation with spin, exchange symmmetry1. The Hamiltonian for the deuteron, a bound-state of a proton and neutron, may be written in theformH=P2p2Mp+P2n2Mn+V1(R) +V2(R)~Sp·~Sn,(1)whereRis the relative radial coordinate. Both are spin-1/2 particles, but they are not identical.(a) The total angular momentum operator is~S=~Sp+~Sn. The state|spsns miis the simultaneouseigenstate of~Sp,~Sn,S2, andSz. What are the allowed values of the total spin quantum numbers? For eachs-value, what are the allowedmquantum numbers.(b) Show that|spsns miis an eigenstate of~Sp·~Sn, and give the corresponding eigenvalue. Hint, usethe fact thatS2= (~Sp+~Sn)·(~Sp+~Sn).(c) Give ten distinct quantum numbers that can be assigned to an eigenstates of thisH. Note thatthis includesspandsn, even though they can never change.
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