# ps9 - Physics 212 – Problem Set 9 – Spring 2010 1...

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Unformatted text preview: Physics 212 – Problem Set 9 – Spring 2010 1. Permutation operator (a) Show that if h Ψ | ˆ O| Ψ i = h Ψ | ˆ P † jk ˆ O ˆ P jk | Ψ i , (1) then h Ψ | ˆ O| Ψ i = h Ψ | ˆ P † jk ˆ O ˆ P jk | Ψ i , (2) where | Ψ i and | Ψ i are each identical particle wavefunctions, ˆ P jk is an interchange operator, and ˆ O is an observable for an identical particle system. (It is possible to show this without making any assumptions about the interchange operator. However, if you do wish to use any properties of the interchange operator, you must prove them first.) (b) Show that if Ψ, representing a many identical particle system, is an eigenvector of the exchange operator ˆ P jk (for all j and k ), then all of the eigenvalues (in that subspace) of ˆ P jk for all j and k are identical. (c) Show, using the permutation operator, that antisymmetric identical particle wavefunctions are orthogonal to symmetric identical particle wavefunctions (d) Show that the operator ˆ S = X P...
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ps9 - Physics 212 – Problem Set 9 – Spring 2010 1...

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