(adsbygoogle = window.adsbygoogle || []).push({}); 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?

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# Homework Help: Time Reversal Invariance Of Hamiltonian

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