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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
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