How do you find the normalized eigenvectors of a hermitian operator in the |n> basis?

Question

O= a |1><1| +a |2><2| + ib |1><2| - ib |2><1|

How do I find the normalized eigenvectors of O in the |n> basis?

Answer 1

Oh, my. A bit high-brow for this forum.

Let's see, suppose |e> is an eigenvector of O that hasn't yet been normalized with eigenvalue e. So,

O|e>=e|e>. and =c where c is a constant.

It's normalized if c=1. Multiplying |e> by a constant is also and eigenvector, so the normalized eigenvector is just |e>/. Operate on this with O as defined above to express it in the |n> basis.

I hope that's all you wanted. You haven't defined |n> so I can't go any further. Also, are you sure you didn't mean orthonormalize? That's more complicated. I suggest getting Shankar's book _Principles of Quantum Mechanics_. I've not see a better math intro to QM, and it's all in bra-ket notation.

Answer 2

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Answer 3

If you're in a class where you need to be able to do this, and you're asking here, in the land of alien abduction physics, well then...

Ask your TA.