1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Time Reversal Invariance Of Hamiltonian

  1. Dec 24, 2008 #1
    1. The problem statement, all variables and given/known data
    Suppose that the Hamiltonian is invariant under time reversal: [H,T] = 0. Show that, nevertheless, an eigenvalue of T is not a conserved quantity.

    2. Relevant equations

    3. The attempt at a solution

    Using Kramer's Theorem.

    Consider the energy eigenvalue equation, [tex]H|\Psi\rangle = E|\Psi\rangle[/tex] for a time-reversal-invariant Hamiltonian, TH = HT. Therefore
    [tex]HT|\Psi\rangle = TH|\Psi\rangle = ET|\Psi\rangle[/tex], so both [tex]|\Psi\rangle[/tex] and [tex]T|\Psi\rangle[/tex] are eigenvectors with energy eigenvalue E.
    This implies two possibilities.

    1. [tex]|\Psi\rangle[/tex] and [tex]T|\Psi\rangle[/tex] are linearly dependent, and so describe the same state, or

    2. They are linearly independent, and so describe two degenerate states.

    It can further be shown that case 1 is not possible in certain circumstances.

    How can I show that there is no conserved quantity?
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted