Hamiltonian which is invariant under time reversal question.

[MathematicalPhysicist]

Homework Statement

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.

Homework Equations

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

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?

 

[Jhero]

Your proof is correct, but here is a more detailed explanation:

First, we know that the Hamiltonian is invariant under time reversal, which means that THT^-1=H. This means that the time-evolution operator, e^{-iHt/\hbar}, is also invariant under time reversal, i.e. Te^{-iHt/\hbar}T^-1=e^{-iHt/\hbar}. This is because the time-evolution operator is given by e^{-iHt/\hbar}=\sum_n c_n e^{-iE_nt/\hbar}|E_n>, where |E_n> are the eigenstates of the Hamiltonian and c_n are some coefficients. Since T commutes with H, it also commutes with e^{-iHt/\hbar}, and therefore T|E_n>=|E_n>. This means that T|psi_0>=|\psi_0>, where |psi_0> is the initial state of the system.

Next, we can rewrite the wave function at time t as \psi(x,t)=<x|e^{-iHt/\hbar}|\psi_0>. Using the fact that T|psi_0>=|\psi_0>, we can rewrite this as \psi(x,t)=<x|Te^{-iHt/\hbar}T^-1|psi_0>. Now, using the fact that Te^{-iHt/\hbar}T^-1=e^{-iHt/\hbar}, we get \psi(x,t)=<x|e^{-iHt/\hbar}|\psi_0>=<x|T^{-1}e^{-iHt/\hbar}T|psi_0>. Finally, using the fact that T^{-1}T=1, we get \psi(x,t)=<x|T^{-1}e^{-iHt/\hbar}T|psi_0>=<x|exp(iHt/\hbar)|psi_0>. This shows that the wave function at time t is equal to the complex conjugate of the wave function at time -t, i.e. \psi(x,t)=\psi^*(x,-t).

Since the wave function at any given instant of time t is equal to the complex conjugate of the wave function at time -t, we can always choose the wave function to be real by choosing t=0. This means