|Mar20-12, 02:17 PM||#1|
Prove Hermitian with two different wave functions
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 Â 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)^* Â 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]
[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.
|Aug6-12, 10:45 AM||#2|
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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