# Do wave functions go to zero at ~ct?

1. Jul 13, 2015

Suppose you have a free election and you make a measurement of its position r_0 at time t = 0. You then wait some time t required for the wave function to evolve out of its collapsed eigenstate. The electron now supposedly has a wave function expanding to infinity in all directions, albeit with exponentially decreasing amplitude. Thus there should be a small probability that we (or I suppose by necessity, someone else) find the electron at some distance d > ct from r_0. Yet we should never be able to find the electron at distances greater than ct from r_0 since its velocity cannot exceed c.

Do wave functions actually reach to infinity, or do they go to zero at ct, or something else?

Last edited: Jul 13, 2015
2. Jul 13, 2015

### maNoFchangE

One thing to note is that a wavefunction is not something that propagates, that is, it's not like in the beginning the value of a wavefunction at some distance z is zero then at some later time our initial wavefunction has propagated to reach z. A wavefunction always extends to infinity at every time.
Let's suppose that our electron is described by a wavepacket. By definition, even at time t=0 where the wavepacket is peaked, let's say at z=0, we still have some probability to find this electron at any position.

3. Jul 13, 2015

### WannabeNewton

Are you asking about propagators in QFT or wave-functions in QM? There is no notion of a light cone in QM so why should the probability amplitude have compact support on light cones?

4. Jul 13, 2015

Wave functions in QM. So I guess you're saying that canonical QM formalism just doesn't apply to this regime, requiring QFT?

I think you misunderstand. My beef is that the electron would appear to have travelled faster than c. The wave function does not propagate, but to know how far the electron travelled, we'd need an initial position measurement, which collapses to a delta function that genuinely is zero everywhere except at the measured position. Thus we'd have to wait for the wave function to evolve out of that eigenstate before a subsequent measurement.

Last edited: Jul 13, 2015
5. Jul 14, 2015

### Orodruin

Staff Emeritus
Since QM builds on classical mechanics and not on relativity, you should not be surprised that this happens. You can have particles travelling faster than c in classical mechanics too, it is just that classical mechanics is not a very good description at relativistic velocities.

6. Jul 14, 2015

### bhobba

To the OP. Read Chapter 3 of Ballentine - Quantum Mechanics - A Modern Development for the full detail of why that's the case.

If you want to incorporate relativity you need Quantum Field Theory.

Thanks
Bill