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## Homework Statement

Given [tex] (\psi_1, \psi_2)=\int dx \psi_1^*(x) \psi_2(x) [/tex], show [tex] (\psi_1, \psi_2)=\int dk \phi_1^*(k) \phi_2(k) [/tex], where [tex] \phi_{1,2}(k)= \int dx \psi_k^*(x) \psi_{1,2}(x) [/tex] and [tex] psi_k(x)=\frac{1}{\sqrt{2 \pi}} e^{ikx} [/tex].

## Homework Equations

[tex] \psi (x)= \int dk \phi(k) \psi_k(x) [/tex]

[tex] \psi(x)=\int dk \phi(k) \psi_k(x) [/tex]

## The Attempt at a Solution

[tex] (\psi_1 , \psi_2)= \int dx \left \{ \int dk \phi_1^*(k) \psi_k^*(x) \right \}\left \{ \int dk \phi_2(k) \psi_k(x) \right \} [/tex]

[tex] =\int dx \left \{ \int dk \phi_1^*(k) \frac{1}{\sqrt {2 \pi}}e^{-ikx} \int dk \phi_2(k) \frac{1}{\sqrt {2 \pi}}e^{ikx} \right \} [/tex]

[tex] = \frac{1}{2 \pi}\int dx \left \{ \int dk \phi_1^*(k) \int dk \phi_2(k) \right \} [/tex]

Is this correct so far? How do I proceed from here? It looks like a Fourier Transform with the 1/2pi. And I have two integrals within another one for the dx. Can I separate them some how?