Time Reversal in Quantum Mechanics:
We wish to define an operation of time reversal, denoted by T, in quantum mechanics. We demand that T be a ``physically acceptable'' transformation, i.e., that transformed states are also elements of the Hilbert space of acceptable wave functions, and that it be consistent with the commutation relations between observables. We also demand that T have the appropriate classical correspondence with the classical time reversal operation.
Consider a system of structureless (``fundamental'') particles and let X=(X1,X2,X3) and P=(P1,P2,P3) be the position and momentum operators (observables) corresponding to one of the particles in the system. The commutation relations are, of course:
[Pm,Xn]=−iδmn,[Pm,Pn]=0,[Xm,Xn]=0.
The time reversal operation T:t→t′=−t, operating on a state vector gives (in Schrödinger picture -- you may consider how to make the equivalent statement in the Heisenberg picture):
T∣ψ(t)⟩=∣ψ′(t′)⟩.
The time reversal of any operator, Q, representing an observable is then:
Q′=TQT−1
(a) By considering the commutation relations above, and the obvious classical correspondence for these operators, show that
TiT−1=−i.
Thus, we conclude that T must contain the complex conjugation operator K:
KzK−1=z∗,
for any complex number z, we require that T on any state yields another state in the Hilbert space. We can argue that (for you to think about) we can write: T=UK, where U is a unitary transformation. If we operate twice on a state with T, then we should restore the original state, up to a phase:
T2=η1,
where η is a pure phase factor (modulus =1).
(b) Prove that η=±1. Hence, T2=±1. Which phase applies in any given physical situation depends on the nature of U, and will turn out to have something to do with spin, as we shall examine in the future.