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here's a problem I am having.

Consider the hilbert space of two-variable complex functions [tex]\psi (x,y)[/tex].

A permutation operator is defined by its action on [tex]\psi (x,y)[/tex] as follows.

[tex]\hat{\pi} \psi (x,y) = \psi (y,x) [/tex]

a) Verify that operator is linear and hermitian.

b) Show that

[tex]\hat{\pi}^2 = \hat{I}[/tex]

find the eigenvalues and show that the eigenfunctions of [tex]\hat{\pi}[/tex] are given by

[tex]\psi_{+} (x,y)= \frac{1}{2}\left[ \psi (x,y) +\psi (y,x) \right] [/tex]

and

[tex]\psi_{-} (x,y)= \frac{1}{2}\left[ \psi (x,y) -\psi (y,x) \right] [/tex]

I could show that the operator is linear and also that its square is unity operator I . I did

find out the eigenvalues too. I am having trouble showing that its hermitian and the

part b about its eigenfunctions.

Any help ?

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# Homework Help: Hermiticity of permutation operator

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