Principles of Quantum Mechanics · Problem 10.3.5

Symmetries and Their Consequences — R. Shankar

Physics — Quantum Mechanics · exchange-operator

Problem

Consider the exchange operator

P_{12}

whose action on the

X

basis is

P_{12}|x_1, x_2\rangle = |x_2, x_1\rangle

((1)) Show that

P_{12}

has eigenvalues

\pm 1

. (It is Hermitian and unitary.) ((2)) Show that its action on the basis ket

|\omega_1, \omega_2\rangle

is also to exchange the labels 1 and 2, and hence that

\mathbb{V}_{S/A}

are its eigenspaces with eigenvalues

\pm 1

. ((3)) Show that

P_{12}X_1P_{12} = X_2

P_{12}X_2P_{12} = X_1

and similarly for

P_1

and

P_2

. Then show that

P_{12}\Omega(X_1, P_1; X_2, P_2)P_{12} = \Omega(X_2, P_2; X_1, P_1)

. [Consider the action on

|x_1, x_2\rangle

|p_1, p_2\rangle

, 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 unaffected under

H \to P_{12}HP_{12}

and

U \to P_{12}UP_{12}

. Given this, show that any eigenstate of

P_{12}

continues to remain an eigenstate with the same eigenvalue as time passes, i.e., elements of

\mathbb{V}_{S/A}

never leave the symmetric or antisymmetric subspaces they start in.

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