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## 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?