Hermitian operators
Hi, this is actually more a mathproblem than a physicsproblem, but I thought I'd post my question here and see if anyone can help me.
So I'm writing an assignment in which I have to define, what is understood by a hermitian operator.
My teacher has told me to definere it as:
<ϕmAϕn> = <ϕnAϕm>* , where lϕn> and lϕm> is the n'th and m'th unit operator.
And using this i then have to proof the more general definition:
<aAb> = <bAa>*, la> and lb> being arbitrary vectors.
I've tried to do so but I have yet not succeeded:
What I've done is to say: Take a hermitian operator. Since it's hermitian it must satisfy:
A = ∑_(m,n)ϕm>amn <ϕn = ∑(m,n)ϕm> anm*<ϕm
Which when dotted with 2 arbitrary vectorsψ> and <φequals to:
<φAψ> = ∑(m,n) <φϕm>amn<ϕnψ> = ∑(m,n) <ϕmφ>*amn <ψϕn>* =
∑(m,n) <ψϕn>* amn <ϕmφ>*
Since A is hermitian this equals to:
∑(m,n) <ψϕn>* amn <ϕmφ>* = ∑(m,n) <ψϕn>*anm*<ϕmφ>* = [∑(m,n) <ψϕn>anm<ϕmφ>]*
Now the proof would work if ϕn>anm<ϕm = ϕm>amn<ϕn, but that's not right is it?
Can anyone help me how to proove this? I know it's simple, but I'm still finding it a little hard.
