How to Determine the Eigenvalues of a Hermitian Operator?

[Faust90]

Homework Statement

I have a hermitian Operator A and a quantum state |Psi>=a|1>+b|2> (so we're an in a two-dim. Hilbert space)
In generally, {|1>,|2>} is not the eigenbasis of the operator A.

I shall now show that the Eigenvaluse of A are the maximal (minimal) expection values <Psi|A|Psi>.

The Attempt at a Solution

I tried to calculate the expectation, which yields to:

|a|^2 A_11 + |b|^2 A_22 +a b* A_12 +b a* A_21

, where A_kl are the matrix elements of the operator A in the given basis of the Hilbert space.
Now I could try to maximize this w.r.t to a and b, under the constraint that a^2 + b^2 =1. Didn't work very well...

Does anyone have an idea?

 

[ShayanJ]

Show us what you did and what went wrong!

 

[Faust90]

Hey Shyan,

thanks for your answer. I tried to find the maximum by using the Lagrange function, so:

L=|a|^2 A_11 + |b|^2 A_22 +a b* A_12 +b a* A_21+lambda(a^2 + b^2 - 1)

Now I got the problem that I don't know how derive L w.r.t c* (conjugated c). I also think that there is a better way to solve this exercise or?

Best regards

 

[ShayanJ]

Hi
The usual way of dealing with such a situation is to take c and c* as two independent variables(which is reasonable since c* is not a differentiable function of c). So you have a function of four independent variables to maximise with the constraint equation aa*+bb*=1.

 

[Faust90]

Thank you very much! :-)