# Eigenvectors of the Hamiltonian

1. Aug 31, 2011

### Ajihood

Hey guys (this is not a HW problem, just general discussion about the solution that is not required for the assignment),

So I am doing this problem where I had to find the eigenvalues and eigenvectors of the Hamiltonian:

H = A*S$_{x}$$^{2}$ + B*S$_{y}$$^{2}$ + C*S$_{z}$$^{2}$.

Easy enough, just basic linear algebra.

However, I want to interpret what the results I get. So I understand the eigenvalues of H represents the energy levels of the system but what physical interpretation should i take to the eigenvectors?

So by finding the eigenvectors, I find a basis that spans the space I am working in. Why is this important to know? I dont want to lose the forest for the trees and just trying to grapple with why the eigenvectors are important or what they physically mean? (eg energy levels have a physical meaning to me, so the eigenvalues make sense but the eigenvectors seem abstract to me). I know I should follow the math in QM but I like to understand the world, not just apply math tricks... Thanks!

2. Aug 31, 2011

### mathfeel

Eigenvectors are state of definite eigenvalue. In the case of Hamiltonian, eigenvectors are states with definite energy. Now, quantum states evolves by the factor $e^{i E t}$ from the Schrodinger's equation. So a state with definite energy evolves by only multiplying a phase factor in front, i.e. A state with definite energy remains a state with definite energy. They do not become "mixed" with other states of other energy (let's suppose states are non-degenerate) as they evolve in time.

That's why eigenvector of H is important and they are called "stationary states" for this reason.

3. Sep 1, 2011

### sweet springs

Hi, Ajihood.

Please let me know what kind of system are you dealing with this Hamiltonian.