Finding eigenvalues and eigenvectors from operator matrix elements

[calcisforlovers]

Homework Statement

QM states of a system are described by linear super positions of two linearly independent state vectors psi1 and psi2. These two states are normalized but are NOT orthogonal to each other. A hermitian operator A actes on the two states in the following way.

A|psi1> = 5|psi1>+3|psi2>
A|psi2> = -3|psi1> - 5|psi2>

Determine the eigenvalues and properly normalized eigenvectors of A.

The Attempt at a Solution

I attempted to introduce a general state |phi> = C1|psi1> + C2|psi2>, act A on |phi> and apply the normalization condition on |phi> to determine C1 and C2 but have gotten nowhere. I have found the value of <psi2|psi1>. Thanks for your help.

 

[Dick]

First find the eigenvalues and eigenvectors of A in the psi1, psi2 basis. The eigenvectors must be orthogonal since the operator is supposed to be hermitian. Now using that psi1 and psi2 are normalized, you should be able to normalize the eigenvectors.