B Speed of electrons in a 2-slit experiment

[jeremyfiennes]

TL;DR Summary

In the 2-slit experiment using electrons, what speed do they travel at?

In the 2-slit experiment using electrons, what speed do they travel at?

 

[Vanadium 50]

Whatever speed you'd like. There's no requirement on speed.

 

[jeremyfiennes]

So the electron-particles can travel at any speed. What about the electron-wave, that causes the interference?

 

[Vanadium 50]

I have no idea what you are talking about with "electron-particles" and "electron-waves" being different things, other than that it is not quantum mechanics.

 

[Nugatory]

So the electron-particles can travel at any speed. What about the electron-wave, that causes the interference?

When you say "the electron is moving at speed $v$" that's just another way of saying "the group velocity of the wave is $v$".

In this problem it is unhelpful and misleading to think of the electron as a small object moving towards the barrier and through the slits. What matters is the probability of an electron detection event happening at any given point on the screen.

 

[jeremyfiennes]

It's a question of the wave-particle duality. Light-as-waves and light-as-particles both travel at the same speed c, so no problem. But electons-as-particles can travel at any speed. My question is: what about the electrons-as-waves that give rise to the interference?

 

[Isaac0427]

It's a question of the wave-particle duality. Light-as-waves and light-as-particles both travel at the same speed c, so no problem. But electons-as-particles can travel at any speed. My question is: what about the electrons-as-waves that give rise to the interference?

That's a major misunderstanding about quantum physics. The wave-particle duality is NOT about something being both a wave and a particle. That, as you have noted, would allow one to separate the electron wave and the electron particle, or the light wave and the light particle.

This is real the wave-particle duality: quantum objects like electrons or photons exhibit some behaviors of classical waves and some behaviors of classical particles. Like classical waves, they exhibit interference are are non-localized prior to measurement; like classical particles, they come in discrete units and are localized upon measurement. Depending on the situation, some properties are more important than others, so the electron can be thought of as either LIKE a wave or LIKE a particle in that specific instance. The key part for your question is about how they are localized-- you are thinking that the electron is partially a classical particle that has a location and a speed. This is, unfortunately, not the case. Prior to measurement (hitting the screen), the electron is not localized; that is, we can only talk about the probability of finding the electron at a certain spot, and this probability is described by a wavefunction, which travels at a certain group velocity. In this case, I would say the electron is behaving more like a classical wave than a classical particle.

So, the only velocity we can talk about is the velocity of the wavefunction-- the velocity of the electron's probability cloud. That is what can go at any velocity. Take a look at this simulation, and click on the single particle tab:
https://phet.colorado.edu/en/simulation/legacy/quantum-wave-interference

 

[vanhees71]

I'd put it more strictly as: Since the discovery of modern quantum mechanics by Born, Jordan, and Heisenberg in 1925 together with Born's probabilitistic interpretation of the quantum state there is no more any wave-particle duality. This intrinsically inconsistent picture of old quantum theory has been overcome with the modern theory, which is consistent.

One should stress that photons are never like point particles. There's no way to localize them at all, because there's not even a position observable you can define for them. This is because the case of massless quantum fields has to be treated separately from the case of massive ones. Massless quantum fields with spin $\geq 1$ do not admit the definition of a position observable in the usual sense and thus should not be interpreted as particles. All you can say about a photon (a Fock state of the em. field with definite photon number 1) is the probability to be detected at place defined by the position of the (of course massive) detector.

For massive fields you can always construct a position operator (no matter which spin), but also here all there is you can define are the probabilities (or probability distribution) for finding a particle at a given place.

 

[Isaac0427]

I'd put it more strictly as: Since the discovery of modern quantum mechanics by Born, Jordan, and Heisenberg in 1925 together with Born's probabilitistic interpretation of the quantum state there is no more any wave-particle duality. This intrinsically inconsistent picture of old quantum theory has been overcome with the modern theory, which is consistent.

One should stress that photons are never like point particles. There's no way to localize them at all, because there's not even a position observable you can define for them. This is because the case of massless quantum fields has to be treated separately from the case of massive ones. Massless quantum fields with spin $\geq 1$ do not admit the definition of a position observable in the usual sense and thus should not be interpreted as particles. All you can say about a photon (a Fock state of the em. field with definite photon number 1) is the probability to be detected at place defined by the position of the (of course massive) detector.

For massive fields you can always construct a position operator (no matter which spin), but also here all there is you can define are the probabilities (or probability distribution) for finding a particle at a given place.

I think the wave-particle duality is still useful to think about, if you think about it in terms of what quantum objects behave like, as opposed to what they are.

You are right that I should clarify: when I say localized, I really mean approximately localized. It is of course impossible to define an exact position for anything, more so with photons. However (correct me if I'm wrong), in the double slit experiment, should one send one photon at a time, the photon could be approximately localized in the sense that we can tell the approximate x-coordinate at which it hits the screen, right?

 

[vanhees71]

Sure, you can think about wave functions (in non-relativsitic QM) in terms of classical waves, because of the mathematics describing them, but physics wise it describes a completely different quantity, namely (via the modulus squared) the probability for finding the particle at a given place at a given time. Similar wave equations also describe sound waves as well as electromagnetic waves. Nobody would think that the corresponding fields have the same meaning in the sense of describing phenomena of Nature.