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# 2 - PHYS 560 Assignment 6 SOLUTIONS Martin J Savage...

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Unformatted text preview: PHYS 560: Assignment 6 : SOLUTIONS Martin J. Savage December 18, 2009 Abstract The sixth assignment for Nuclear Physics , PHY560, Autumn 2009. December 2009 1 Assignment 6 : PHYS 560 Due : Dec 14 1. Wong : 6.1 : We wish to construct the allowed values of total angular momentum for a system composed of two phonons each with angular momentum λ . The wavefunction must be totally symmetric under interchange. If we start by considering the stretch state, M = 2 λ , then (a) There is only one way to obtain this : | λ i ⊗ | λ i . This state is allowed as it is symmetric under interchange, and is the stretch state associated with J = 2 λ . (b) For the state with M = 2 λ- 1, we have states | λ i ⊗ | λ- 1 i and | λ- 1 i⊗| λ i . The sum of these states gives the M = 2 λ- 1 state of the J = 2 λ multiplet. (c) For the state with M = 2 λ- 2, we have states | λ i ⊗ | λ- 2 i , | λ- 2 i ⊗ | λ i , | λ- 1 i ⊗ | λ- 1 i . The linear combination ∝ | λ i ⊗ | λ- 2 i + | λ- 2 i ⊗ | λ i + 2 | λ- 1 i ⊗ | λ- 1 i is the M = 2 λ- 2 member of the J = 2 λ multiplet as is obvious from acting with the lowering operator on the M = 2 λ- 1 state), while the combination ∝ | λ i⊗| λ- 2 i + | λ- 2 i⊗| λ i-| λ- 1 i⊗| λ- 1 i is the M = 2 λ- 2 member of the J = 2 λ- 2. (d) This pattern continues all the way through the spectrum. There- fore there are no states with odd J in the state formed from two identical bosons each with λ . For the state composed of three λ = 2 bosons, we simply construct all possible states. I will shorthand this, and write states with sym- metrization implicit. The number of states with different values of the m i corresponds to the number of symmetrized, linearly independent states–hence states of the systems: M = 6 : Denoting the tensor product of states by | m 1 , m 2 , m 3 i , we have | 2 , 2 , 2 i , and hence only one state with J = 6. M = 5 : | 2 , 2 , 1 i , hence only one state, there is NO J = 5 state. M = 4 : | 2 , 2 , i , | 2 , 1 , 1 i , hence there are two states, there is J = 4 state. M = 3 : | 2 , 2 ,- 1 i , | 2 , 1 , i , | 1 , 1 , 1 i , hence there are three states, there is J = 3 state. 2 M = 2 : | 2 , 2 ,- 2 i , | 2 , 1 ,- 1 i , | 2 , , i , | 1 , 1 , i , hence there are four states, there is J = 2 state....
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2 - PHYS 560 Assignment 6 SOLUTIONS Martin J Savage...

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