Prove Hermitian with two different wave functions

pfollansbee

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

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

2. Relevant equations

[tex]∫ψ(r)^*  ψ(r)dr = ∫Â^* ψ(r)^* ψ(r)dr[/tex]

3. The attempt at a solution

It is obvious to me that if
[tex] <m|\hat A|n> = <\hat A m|n>[/tex]
then
[tex] <m|\hat A|n> = <n|\hat A|m>^*[/tex]

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
[tex]\phi _m = (\psi -c_n \phi _n)/c_m[/tex]
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.

pfollansbee

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.

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