# Hamiltonian which is invariant under time reversal question.

1. Nov 5, 2009

### MathematicalPhysicist

1. The problem statement, all variables and given/known data
Assuming that the Hamiltonian is invariant under time reversal, prove that the wave function for a spinless nondegnerate system at any given instant of time can always be chosen to be real.

2. Relevant equations
$$\psi(x,t)=<x|e^{-iHt/\hbar}|\psi_0>$$
The Time-Reversal operator: $$T^2=-1 \\ T|x>=|x>$$
HT=TH.

3. The attempt at a solution
$$\psi(x,t)=<x|e^{-iHt/\hbar}|\psi_0>=<x|e^{-iHt/\hbar}TT^-1|\psi_0>=<x|Te^{iHt/\hbar}T^{-1}|\psi_0>=<x'|exp(-iHt/\hbar}|\psi _0'>=<x|Texp(-iHt/\hbar)T^-1|\psi_0>=(1)$$
$$(1)=<x|exp(iHt/\hbar)|\psi_0>=\psi(x,t)*$$
where the last line is assured because |x'>=T|x>=|x>, |\psi_0'>=T|psi_0>.
Is this fine as a proof or not?