(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I am given that an operator is defined by the following:

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

in a 2 variable Hilbert space of functions [tex]\psi(x,y)[/tex]

I have to show that the operator is linear and Hermitian

2. Relevant equations

An operator is self adjoint if:

[tex]\left\langle\psi\right|\hat{A}^{\dagger}\left|\right\phi\rangle=\left\langle\phi\right|\hat{A}\left|\right\psi\rangle^{*}[/tex]

3. The attempt at a solution

With the information given, how can I apply the above condition? Should I put x one side and y on the other? Or should I use psi(x,y) on each side. If so, does the fact that one of the psi's is a bra and the other is a ket change anything?

To show that it is linear, I need to show that it is distributive. Is it as simple as saying:

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

If I remember correctly, my professor said that since the space is linear then the operators are distributive. That doesn't make sense to me. If an operator can be anything, then couldn't it be something that isn't linear as well?

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# Self Adjoint Operator

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