I don't usually consider "waves" to have mass. They're just energy that moves THROUGH mass. Light waves, sound waves, ocean waves... They are all massless energy. But I'm thinking about quantum wave/particle dualities. Electrons have mass and I'm having some trouble accepting how waves can have mass. Any insights or ideas for this seemingly trivial road-block?
Matterwaves, as they are called, aren't physical waves like sound, nor are they EM waves like light...they are probability waves. The probability wave itself does not have any "mass", it just tells you the probabilities of finding the particles at specific places. The particle has mass, not the wave. The wave simply describes the particle.
How funny! You're more famous than I though, Matterwave. So, you're saying that it's not just a "wave/partice" duality when it comes to electrons, it's also a "mass/massless" duality? So when an electron hits the film in the double slit experiment, it records mass when it's a particle and no mass when it's a wave?
Err not exactly. The wave/particle duality is a bit of a different subject where the electron( or particle ) sometimes acts like a wave and sometimes acts like a particle. The wave equation was developed when we started looking at the idea that all matter is made up of waves, however there are still particle like aspects. So now we have these waves with no medium that are peculiar in that the entire wave seems to collapse when it is measured at any one position. The wave equations give scientists all the information they need to know about a particle even though there is no physical explanation for it. In some ways it is like a computer program trying to figure out what makes its bits flip.
Hmm, I think we should be very careful when discussing wave/particle duality and what exactly this means. Wave-particle duality generally means that the electron acts like a particle for some experiments and like a wave for others. This means, in simplified terms, if I am looking for particle-properties such as mass, the electron will act as a particle. If I am looking for wave-like properties such as diffraction, the electron will act like a wave. If you try to measure the mass of the electron, you get a definite mass because you are looking for a particle-tied property and the electron will act as a particle for this test! You will never measure the electron to have zero mass, because the electron will NOT act like a wave for a mass measurement. I'm trying to keep this discussion close to High-school level, so at higher levels of understanding, the picture is more complicated. I will digress a little bit into the more complicated picture, but if you don't understand it at this point, don't worry. You will, once you study QM at a deeper level. The wave is, as I mentioned, a probability wave. The particle is described by a wave-function, and this wave-function is NOT physical in any sense. You can't make any measurements on this wave-function. So it doesn't make sense to try to measure the "mass" of this wave-function. All you can do is measure many electrons prepared in the same state to try to get a feel for the probability distribution of the electrons. The double slit experiment, for example if you release one electron at a time, each detection event is particle-like. You see 1 electron at one detector, and then 1 electron at another detector. You never detect some sort of "wave". Where the wave characteristics come in is when you get many detections, you will see a diffraction pattern IN your detections which would not arise classically for particles. It is therefore easier to describe the sum total diffraction phenomenon in terms of waves.
Sure, but when you make measurements in tunneling, you always measure the electron as either having tunneled or not having tunneled. You never measure an electron as "half-tunneled" or some such. Only when you get many electrons together, can you find that X% of them tunneled and 100-X% of them didn't. There is a definite disconnect between how electrons are described and how electrons are measured. They may be described by a wave-function, but they will inevitably be measured as electron particles.
Another example of the wave-particle duality. I wouldn't expect you would need many electrons though. You could setup an experiment so that the potential allowed for a 90% chance of tunneling even though V > p_e. Do one experiment and if the single electron tunnels then you just proved wave like nature without any statistics. If it happens not to tunnel then wait 5 years, well you get the point.
? First of all, how does 1 data point make any proof? Second of all, you are STILL detecting the electron AS AN electron. Perhaps I am not understanding your scenario correctly...
Just imagine you did an experiment once and got "lucky" and the electron tunneled. The wave equation describes the result and the "wave" is not based of a statistical collection of results.
This is interesting, and you haven't lost me at all. You're definitely bringing to light some simple, missing pieces in laymans literature. Let me pick through some of this for clarification. Are you saying that a single electron will never display a "wave" pattern, even if we try to measure it as such? Are you also saying that an electron cannot tunnel unless it is in a group?
No, he is saying that the wave nature described in QM is based off a statistical collection of results. We can never say exactly what single particle will do, just what many of them will do, though in the end if 5 million electrons tunneled, they each tunneled. As for your first question, I think that a single electron can display wave nature when measured only once. MatterWave might have more to say on that though.
You can find the electron within the area of space where V > p_e though. The probability wave is a gaussian here. Not sure thats what you meant though.
So you are saying, by virtue of the electron having tunneled, it must have wave-characteristics because particle characteristics cannot account for tunneling? Perhaps I should make my point clearer. When you make a measurement, e.g. a measurement of mass, you invariable are measuring the particle. Even in the tunneling example, you are measuring the position of the particle. You are not measuring a wave like you could measure a physical wave (like waves on a pond). With waves on a pond, I could measure the amplitude by using a ruler, for example; however, I can make no such measurements on the electron's wave-function (or the absolute square of the wave-function). I can only get an idea for the amplitude, if I get many measurements of identically prepared electrons. With your tunneling example, with 1 electron, I certainly can't measure the amplitude, or the wavelength, or any other wave-characteristic of the wave-function. All that I may be able to tell is that something weird is happening in that the electron moved through a region it shouldn't be able to move through. Perhaps I am wrong. And indeed, it is often hard to reconcile high-school level explanations with "real" explanations. But in any case, this is a digression from the OP's discussion. I think the main point as far as the OP is concerned is that if I measure the mass of an electron I will always measure a mass because the electron behaves particle-like for such a measurement.
Actually you can never find the electron in a classically forbidden region (V>E). You can only find it on one side or the other. The rough proof is that if you tried to measure the electron in the classically forbidden region, you must necessarily boost the energy of the electron such that that region is now classically allowed. You can't tell that originally the electron didn't have enough energy. This is intricately intertwined with the HUP. I don't have a very good proof, but there are rigorous proofs available, and you can search for them. =)
I agree with everything you say MatterWave, but not sure how it contradicts the fact that there are wavelike-characteristics with emphasis on "like". I am not saying it disproves the particle theory, just that there is wavelike nature describing the experiment. Just trying to say that wave equations are not all based off statistical results, otherwise it would be a boring subject and talk of a single particle borrowing energy, or interfering with itself would no longer be a subject.
I appreciate your input, LostConjugate. I hope my continued questioning doesn't bother you. I do have a few layman's books on the subject and I feel pretty comfortable with things like probability distributions. I know that, in a non-controlled environment, particles will do (essentially) whatever they want without us being able to track exact locations or states (uncertainty principle). I also know that, in more natural environments, many electrons are required to "fulfill" a probability distribution. Without enough electrons, it's a free-for-all. I think even WITH enough particles it's still a free-for-all, but we only care how they perform as a group. I think I'm most interested in controlled experiments, like the double-slit one. What I'm picking up here is that the wavelike properties of electrons are described from the group that collectively display the wave pattern. Is that right? But a single photon CAN display both wave and particle dualities. Is this right?
A photon and electron both display wave particle dualities the same. A photon is a particle, it is not an electro magnetic wave. So with the double slit you can work with either one, shouldn't matter. Either way in order to see the wave nature you must send more than one particle through the slits. Though you can send them with great amounts of time between. Like 1 photon every year and still see the interference pattern.
Discussing this topic can become quite sticky. I myself prefer the Ensemble interpretation of QM...but in discussing this topic I tried to not allow interpretation-dependent aspects through, and just describe standard quantum mechanics. It's difficult because the topic we are discussing is beginning to push onto the interpretation level...haha. Let us try to make everything as clear as possible. First, let's stay with the double-slit experiment. When you make a specific measurement of position, i.e. where the particle is on the screen, you necessarily measure only the particle-like aspect - position (waves are spread out!). When you combine many many measurements of position, you start to see a diffraction pattern occur in the distribution of these electrons and from this you can infer a wave-like property. Be absolutely certain that you did not measure this wave-like property, but you inferred it from your measurements. This applies to both electrons and photons. You can ever only measure 1 aspect at a time. Now onto the tunneling example. In fact, it is analogous because when you measure the 1 electron, you are in fact measuring a particle-like aspect (position). You may be able to infer from this 1 electron the wave-nature but you certainly did not measure the wave-nature. I admit, this topic is getting to the limits of my QM knowledge. The main point to make here is that the wave-function is not a physical reality (except perhaps in the Many-worlds interpretation....which I'm not familiar with). The wave-function cannot be measured like position or mass or the amplitude of a real physical wave.