# Probability wave speed of dispersion and interference

I'm a layperson learning about quantum mechanics and probability waves. My understanding is that the probability wave for the position of a particle disperses throughout all of the universe.

I have two questions about probability waves:

1) Does the probability wave for a particle take time to disperse throughout space?

For example, if a particle is formed at $t=0$ and I have a detector one light year away, will it be impossible for the detector to detect the particle until the probability wave reaches the detector and at least establishes probability where the detector is located? If so, is there a given speed for the wave (e.g., speed of light)?

2) In the double split experiment where single electrons are fired and detected on the other side of the slits, why do the physical barriers cause the waves to interfere with each other? I was thinking that if the particles were photons, then would the double slits cause the same interference? What if the slits were cut into perfectly clear glass? I assume that the probability waves would not be disrupted by the glass, but is that true?

1) Yes. Bearing in mind Heisenberg's Uncertainty Principle, one must note that's impossible to correctly distinguish the actual speed and position of a particle or wave at the same time. It's really only theoretical that the speed of light is the limit of motion, but that's what tend to assume.

2) The interference bands of the double slit experiment are mainly due to phase cancellation when applied to light waves and sound waves. The difference in phasing is mainly caused by the slight differences in the time for the wave's contents to hit remaining slits or backing wall. Replacing the opaque walls with (flawless) glass might cause distortion in a similar band pattern (again, due to the time it takes the photons to get through the glass), but I doubt it'd be that noticeable by the naked eye. Photovoltaic cells would show the result better, with less light absorption over time where you'd normally find the bands.

• Thank you very much. I didn't word question 2 well. When a single photon is fired in the double slit experiment, it initially has a single probability wave, correct? When the wave hits the slits, it appears that the wave splits into two waves that interfere with each other. The wave can't travel intact through the barrier surrounding the slits. Why would that be? And, it appears that the probability wave would behave differently if the barrier surrounding the slit were opaque or clear. If the photon has no position until observed, why is it that the material impacts its probability wave? – user14336 Oct 25 '12 at 2:52
• This is the wrong answer to question 1--- the propagation is instantaneous even in relativity theory, you need to take into account particle/antiparticle formation at your detector to restore causality. – Ron Maimon Oct 25 '12 at 3:32

For your first question, if you produce a particle which is localized at a given point, it instantly spreads to fill all space, and there is no speed limit. This is a nonrelativistic property--- the nonrelativistic propagator is of constant magnitude for a delta-function initial condition (infinitely concentrated particle), and this is intuitive, since the momentum is infinitely uncertain.

If you make the initial condition be a wavepacket, the wavepacket width spreads as the square root of time. This is more reasonable, but there is still an amplitude for moving as fast as you like, since you can't make the particle stay in a finite region.

For relativistic particles, the result is much the same, although with slightly different propagation functions. If you make a localized electron, you will find an amplitude for it to move faster than light, which goes to zero away from the light-cone with an e-folding length which is roughly the Compton wavelength of the electron.

This doesn't violate causality only because there is such a thing as an electron positron pair creation. If you create a detector to measure the superluminal propagation of the electron you made, it will not be possible to disentangle the faster-than-light propagation of a single electron from detecting a different electron pair-produced from the vacuum. This is the famous reason that relativistic theories need antiparticles, and it was understood (to various degrees) by Dirac, Schwinger, Feynman and Dyson.

For the second question, if the glass has an index of refraction, you will still see interference fringes. But glass with an index can't be perfectly transparent. What you can do is make an index smooth rise, so that the index of refraction gradually rises in the glass then gradually falls, and in this case, the light going through the holes will just be out of phase with the light going through the glass. This out-of-phase light will spread and interfere according to Huygens-Fresnel principle, and will make an interference pattern as always.

• Thanks very much. I appreciate the answers. I understand that a probability wave isn't something that can be detected, only that its effects can be observed. Still, it seems that a probability wave has properties attributed to it that cause it to interact with matter (e.g., with the first barrier). And, the properties attributed depend on the type of particle (e.g., an electron's probability wave is affected by glass differently than a photon's probability wave). How can this be? Or, am I wrong and that the interaction of probability waves of different particles with matter are identical? – user14336 Oct 25 '12 at 16:29
• @user14336: It is good to separate the "probability wave" from the "matter wave". The "probability wave" is the wavefunction, and it's a property of the whole system, of all the particles, not of any one. The "matter wave" is the field associated with each particle type. The distinction is not made often explicitly, and this causes confusion. The field of an electron makes a different kind of wave, and it interacts differently with the other fields. It's best to analyze many examples of an electron in a potential, where you have nonrelativistic QM. Photons are always relativistic. – Ron Maimon Oct 25 '12 at 17:28