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I'm doing quantum mechanics with only a little experience in linear algebra. I've been working on eigenstates/values/functions/whatever for a couple days but still having a little trouble. Here's a question I had recently, and if anyone can do a quick check of my work and point me in the right direction, I'd appreciate it.

The Hamiltonian operator for a system in 2-d vector space is iΔ(|w

The problem asks me to do a bunch of things:

1) Represent H(hat) and Ω(hat) in a matrix in the basis |w

2) Find the eigenvalues E

3) Express |w

4) Finally, use this linear combination to express H(hat) and Ω(hat) in the basis |E

Now, first I represented H(hat) as a matrix that looked like 0, iΔ (row 1) and -iΔ, 0 (row 2). I wasn't 100% sure how to do Ω(hat), but I figured the only non-zero values will be on the main diagonal, since we're representing it in the basis of its eigenstates.

Then, I solved for the eigenvalues and got E

In terms of making these eigenstates linear combinations of each other, I'm sure that means transform |w

So, I suppose if anyone can provide a straightforward method to transforming an eigenvalue into an eigenfunction and normalize it, it would be a huge help. Thanks for reading, and for all the help in the past.

EDIT: I guess this wasn't really a "quick" question in the end, sorry. But I feel like most of the helpers on PF can probably shed some light on this in about a sentence or two.

## Homework Statement

The Hamiltonian operator for a system in 2-d vector space is iΔ(|w

_{1}><w_{2}| - |w_{2}><w_{1}|). |w_{1}> and |w_{2}> are eigenstates of an observable operator Ω(hat).## Homework Equations

The problem asks me to do a bunch of things:

1) Represent H(hat) and Ω(hat) in a matrix in the basis |w

_{1}>,|w_{2}>.2) Find the eigenvalues E

_{1}and E_{2}and the normalized eigenstates |E_{1}> and |E_{2}>.3) Express |w

_{1}> and |w_{2}> as linear combinations of |E_{1}> and |E_{2}>.4) Finally, use this linear combination to express H(hat) and Ω(hat) in the basis |E

_{1}>,|E_{2}>.## The Attempt at a Solution

Now, first I represented H(hat) as a matrix that looked like 0, iΔ (row 1) and -iΔ, 0 (row 2). I wasn't 100% sure how to do Ω(hat), but I figured the only non-zero values will be on the main diagonal, since we're representing it in the basis of its eigenstates.

Then, I solved for the eigenvalues and got E

_{1}= -Δsqrt(2) and E_{2}= Δsqrt(2). This is the first area I got stuck -- I sort of "guessed" on how to get the eigen*function*from the eigenvalues. I think it involves solving a system of equations, but again, I'm pretty inexperienced with linear algebra.In terms of making these eigenstates linear combinations of each other, I'm sure that means transform |w

_{1}> and |w_{2}> into an additive combination of the new normalized eigenfunctions |E_{1}>, |E_{2}>. As long as I have |E_{1}> and |E_{2}> this seems straightforward enough.So, I suppose if anyone can provide a straightforward method to transforming an eigenvalue into an eigenfunction and normalize it, it would be a huge help. Thanks for reading, and for all the help in the past.

EDIT: I guess this wasn't really a "quick" question in the end, sorry. But I feel like most of the helpers on PF can probably shed some light on this in about a sentence or two.

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