Introduction: Running Time Backward
Time reversal asks: if we reverse all momenta and run the movie backward, do we get valid physics? In quantum mechanics, time reversal is unusual—it involves complex conjugation, making it an antiunitary operator.
The Time Reversal Operator
Time reversal \(\hat{T}\) is antiunitary:
\[\hat{T}(c|\psi\rangle) = c^*\hat{T}|\psi\rangle\]For spinless particles: \(\hat{T}\psi(x) = \psi^*(x)\)
Under time reversal:
- \(\hat{T}\hat{x}\hat{T}^{-1} = \hat{x}\) (position unchanged)
- \(\hat{T}\hat{p}\hat{T}^{-1} = -\hat{p}\) (momentum reversed)
- \(\hat{T}\hat{L}\hat{T}^{-1} = -\hat{L}\) (angular momentum reversed)
Kramers Degeneracy
For half-integer spin systems with time-reversal symmetry, every energy level is at least doubly degenerate. This is Kramers degeneracy.
Worked Example
The Schrödinger equation \(i\hbar\partial_t\psi = \hat{H}\psi\) becomes under \(t \to -t\):
\[-i\hbar\partial_t\psi' = \hat{H}\psi'\]Complex conjugating both sides restores the original form if \(\hat{H} = \hat{H}^*\). Real potentials are time-reversal symmetric.
The Quantum Connection
Time reversal symmetry is violated by magnetic fields and certain weak interactions. The combination CPT (charge conjugation × parity × time reversal) is believed to be an exact symmetry of nature. Time reversal explains why interference patterns are stable—time-reversed paths contribute equally.