## Prove Hermitian with two different wave functions

1. The problem statement, all variables and given/known data

Let $$ψ(r)= c_n ϕ_n (r) + c_m ϕ_m (r)$$ where $$ϕ_n(r)$$ and $$ϕ_m (r)$$ are independent functions.
Show that the condition that Â is Hermitian leads to
$$∫ψ_m (r)^* Âψ_n (r)dr = ∫Â^* ψ_m (r)^* ψ_n (r)dr$$

2. Relevant equations

$$∫ψ(r)^* Â ψ(r)dr = ∫Â^* ψ(r)^* ψ(r)dr$$

3. The attempt at a solution

It is obvious to me that if
$$<m|\hat A|n> = <\hat A m|n>$$
then
$$<m|\hat A|n> = <n|\hat A|m>^*$$

My professor gave me a hint and said that I need to expand these out and show that they are equal. This is where my problem lies. I have no idea how to expand these out. I have tried a few ways, like setting
$$\phi _m = (\psi -c_n \phi _n)/c_m$$
This certainly did not seem like the correct approach to me.

Maybe someone here can give me another hint as to how this goes. I have asked my professor three times to talk to me about it, but he seems content in misunderstanding me and talking about other problems that we have already solved.

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 Hmm no replies... oh well. Here is the solution that I came up with. Just in case anyone else happens to happen upon a similar problem, this may help. Attached Thumbnails