# Homework Help: Time Reversal Invariance Of Hamiltonian

1. Dec 24, 2008

### casdan1

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

1. $$|\Psi\rangle$$ and $$T|\Psi\rangle$$ 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?