Do wave functions go to zero at ~ct?

[Cadaei]

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?

 

[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.

Yet we should never be able to find the electron at distances greater than ct from r_0 since its velocity cannot exceed c.

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.

 

[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?

 

[Cadaei]

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?

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

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.

I think you misunderstand. My beef is that the electron would appear to have traveled 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.

 

[Orodruin]

I think you misunderstand. My beef is that the electron would appear to have traveled faster than c.

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

 

[bhobba]

Since QM builds on classical mechanics and not on relativity

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