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Eigenvectors of the Hamiltonian

  1. Aug 31, 2011 #1
    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[itex]_{x}[/itex][itex]^{2}[/itex] + B*S[itex]_{y}[/itex][itex]^{2}[/itex] + C*S[itex]_{z}[/itex][itex]^{2}[/itex].

    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. jcsd
  3. Aug 31, 2011 #2
    Eigenvectors are state of definite eigenvalue. In the case of Hamiltonian, eigenvectors are states with definite energy. Now, quantum states evolves by the factor [itex]e^{i E t}[/itex] 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.
     
  4. Sep 1, 2011 #3
    Hi, Ajihood.

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

    Thank you in advance.
     
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