Exercise 10.3.5.* Consider the exchange operator P12 whose action on the X basis is P12/x1, x2) = [X2, Xı) (1) Show that

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Exercise 10 3 5 Consider The Exchange Operator P12 Whose Action On The X Basis Is P12 X1 X2 X2 Xi 1 Show That 1 (88.79 KiB) Viewed 43 times

Exercise 10.3.5.* Consider the exchange operator P12 whose action on the X basis is P12/x1, x2) = [X2, Xı) (1) Show that P12 has eigenvalues #1. (It is Hermitian and unitary.) (2) Show that its action on the basis ket 01, @2) is also to exchange the labels 1 and 2, and hence that Vs A are its eigenspaces with eigenvalues #1. (3) Show that P12X,P12=X2, P12X2P12=X, and similarly for P, and P2. Then show that P1222(X1, P.; X2, P2)P12 =12(X2, P2;X1, Pi). [Consider the action on 1x1, x2) or | P1, P2). As for the functions of X and P, assume they are given by power series and consider any term in the series. If you need help, peek into the discussion leading to Eq. (11.2.22).] (4) Show that the Hamiltonian and propagator for two identical particles are left unaffected under HP12HP 12 and U-P12UP12. Given this, show that any eigenstate of P12 continues to remain an eigenstate with the same eigenvalue as time passes, i.e., elements of Vs/A never leave the symmetric or antisymmetric subspaces they start in.