How to time-reverse a quantum system

Time-reversing a classical Newtonian trajectory is simple. If q(t) and p(t) are the positions and momenta of a particle on the trajectory, then time reversal flips that trajectory as follows:

• q(t) → q(-t) = q(t)
• p(t) → p(-t) = -p(t)
  

For example, a particle traveling along some path with velocity to the left becomes a particle traveling along that path with velocity to the right -- just like when we play a movie in reverse. Very simple.

In quantum mechanics, time-reversal looks comparatively strange, because it involves complex conjugation. (More precisely, it is implemented by an antiunitary Hilbert space operator.) Why? Here's my answer today (more answers later):

One reason that time reversal in quantum mechanics requires complex conjugation is that time-reversing a wave function requires time-reversing its phase.

I recently pointed out an oversimplified way to visualize certain wave functions. Here's a better way to put the idea: The phase of a simple plane wave can be visualized as the assignment of a "dial value" to little regions in spacetime.

In particular, if the plane wave (in the position representation) has the form:

ψ(x, t) = exp(ipx - it),

then exp(-it) -- the phase -- is just a point on a circle, lying on the complex plane. So, we can think of ψ(x, t) as assigning dial-values to points -- each corresponding to a different location on the circle. Moving smoothly forward through time gives rise to a changing dial value. This just represents the changing phase as the plane wave propagates through space:

(Click to enlarge)

Now, how should we time-reverse such a system? Well, minimally, it seems we'd want our dial to run in reverse. That's exactly what conjugation does. Notice that by sending ψ to its conjugate ψ*, we flip the arrow about the real axis of the dial:

The result is an arrow moving in the opposite direction. Moreover, the arrow cannot be reversed by any symmetry operator that does not conjugate (i.e., a unitary operator). That's because the wavefunction ψ(x, t) is given by an inner product, and unitary operators preserve inner products.

So there you have it: one way to see why quantum time-reversal requires conjugation.

As a final note: a wave's phase really just describes its relationship to the origin of a coordinate system. So, one might complain that phase isn't a physical feature of a wave, any more than a coordinate system is. However, differences in phase, and in particular changes in phase, are physical features of a wave. So, our account of time reversal must be sure to reverse these quantities as well.