Applying time reversal to the free particle wavefunction

[Hypersquare]

Hi there!

I have tried to apply time reversal (which makes t -> -t) to a free particle wavefunction:

Exp[i(p.r-Et)/$\hbar$]

and got:

Exp[-i(p.r-Et)/$\hbar$]

I got this by flipping the sign of p since it has a d/dt part, and flipping the t in the Et part. However I think this is wrong since in B.R. Martin's Nuclear and particle physics (page 12) he gets:

Exp[-i(p.r+Et)/$\hbar$]

What have I missed?

 

[TSny]

Hi Hypersquare. We would like the time reversed wavefunction to represent another free particle with opposite momentum but with the same energy. Apply the momentum and energy operators to your function and see if it has both of these properties. Then try the function given in Martin's book.

 

[Hypersquare]

Thanks TSny. I did both of those and it seems that BR Martin was correct! My energy comes out negative and his positive. The solutions agree on application of the momentum operator though. This means that I have misunderstood something entirely then. I am not sure what though.

 

[TSny]

The key point is that just replacing t by -t in a wavefunction does not yield the wavefunction that would describe the "time-reversed" state. For spinless particles, you get the time-reversed state by replacing t by -t and also taking the complex conjugate of the wavefunction. You can check this for Martin's example. For particles with spin, it's more complicated.

If you inspect the Schrödinger equation, you can see that if ψ(r, t) satisfies the equation, then ψ(r, -t) does not satisfy the equation, whereas ψ^*(r,-t) does. (This assumes a Hamiltonian which is invariant under time reversal.)

In general, a time reversal operator in quantum mechanics is not a linear operator, but an "antilinear" operator. This is discussed in some of the standard texts.