Deja Vu (March 2009)
It all started innocently enough. I had been auditing the upper division Quantum Mechanics course at CSU to try to bring my QM skills up so I could get better at Quantum Computing. It was fairly relaxed; I wasn’t doing any of the homework, but I took detailed notes in TeX, complete with formatting the equations on the fly. Mark Bradley, the teacher, was a dynamo at the blackboard though, so it was often a struggle to keep up with his rapidly scribbled equations. So I have very pretty but sometimes incomplete notes.
And then one day in late March, I noticed something a little odd. Or rather, I noticed something that should have been familiar and comfortable and old hat to me, only now it seemed odd and suggestive. The electron wave functions in stationary solutions to the Schrodinger equation . . . you know, things like ”atomic orbitals” . . . have a phase oscillation. Their spatial part doesn’t change – the probability of ”finding the electron at location x” is constant over time – but the complex phase oscillates, and at a rate which is dependent on energy. (”Complex” here meaning ”using imaginary numbers”, not ”complicated”.) Higher energy levels rotate their phase faster than lower ones.
That seemed familiar, but I had a hard time figuring out why. It wasn’t just because I had seen it before, in quantum mechanics and quantum chemistry and quantum computing. Rather, it reminded me of something else, but I had no idea what.
After a couple of weeks of frustration, I finally remembered what it reminded me of: gravitational time dilation in General Relativity. There, clocks that are higher up in a gravitational potential run faster than ones lower down. Now, one of the biggest outstanding problems in modern physics is trying to reconcile General Relativity with Quantum Mechanics. Many people have tried and failed; it’s considered to be a very hard problem. But here we had a case of GR and QM saying similar things. Maybe quantum gravity wasn’t as hard as everyone thought? Maybe this was an easy way in?
So, as an exercise, I decided to see if I could discover the mathematical relationship between those two things. Might they be connected in some way? That would be interesting.
The only real problem was that they are expressed in different terms. One is a phase rotation rate in terms of energy. The other is a time shift in terms of potential. But it’s not too hard to work that through; it only takes high school algebra. One can easily convert the phase into time by using the normal oscillation frequency. And one can easily get energy equivalents for the potential by assuming a ”test particle” with small (but arbitrary) mass m.
In GR the answer is that the rate of time flow is proportional to the total energy (potential energy plus rest mass energy). The time dilation seen for observer b by observer a is just Tb/Ta = Eb/Ea, where Ea and Eb are the total energy of the test particle.
In QM the answer is that the relative rate of phase oscillation (= time flow) is just Tb/Ta = Eb/Ea, where Ea and Eb are the total energy of the electron (including its rest mass).
In other words, charged particles in an electric potential appear to be time-dilated in exactly the same way as massive particles in a gravitational potential. The equations are identical, if you consider the particle’s phase frequency to be its “local clock”.
There followed much thought, crunching of equations, analysis of experiments, and searching the literature. For a few weeks, giddy, I thought I was the first human ever to discover this.
Until I found out I wasn’t.
Energy and Time 