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Wave packet confusion

  1. Oct 1, 2015 #1
    I have a few general questions about how wave packets relate to particle creation and motion.

    1) In the following video, Steve presents us with a simple case of a free particle at rest, where we have zero momentum and thus know the momentum exactly, but as a consequence we have no idea what the position the particle is in, so the probability density is constant in space. So the wave function for the particle only includes the time component.



    Now we take this same particle/wave function and set it into motion relative to a stationary observer or lab frame by introducing a Lorentz transform into the previous time-dependent wave function (see 6:00). The effect of this is to give the particle a wavelength and, thus, a momentum in a left or right direction, and thus a positional dependence added to the wave function. My question here is, "What is the particle we are talking about and where is the particle?" The visualization just looks to me like a continuously oscillating probability density. Does this just mean that the probability of finding a particle oscillates through all space with that sort of distribution? And that that distribution is moving to the right or left? I'm confused.

    2) What actually is the particle? Let's forget about about bosons for the moment and just focus on fermions. In terms of a wave packet, what constitutes the particle? Does a particle, say electron, have to take the form of a standing wave? Is this what the Dirac equation is telling us? The mass that characterizes the electron is made up of both a coupled left moving wave function and a right moving wave function. Does the Dirac equation describe a standing wave? If that's the case, then what is Steve talking about in the above video when he adds momentum to the free particle and says it is now representative of a particle moving to the right? I don't see a standing wave there.

    Does the standing wave define the embodiment of the particle itself, or does it just define a "pocket" of space-time where an otherwise ambiguous entity called a particle is highly likely to be "found." These distinctions or lack thereof are not very clear to me.

    3) Dispersion--When we trap a particle in a potential well, we quantize it's energy, momentum, and position because the "waviness" of the wave packet or wave function must fit integer wavelengths in the well. It seems to me that the "shape" the wave function takes within any given potential energy constraint governs the identity of that particle in the sense that it associates it with a standing wave that has definite energy, say. But what about about a "free" particle? What governs it's identity if it's not under any potential constraint that might shape it? From JMGerton's video here, it would just seem to disintegrate due to "dispersion" as it travels down the road. What does that mean?



    In any case, I'm hoping someone can help clear up some of these quandaries for me in a somewhat simplified fashion . Please notice the "B" prefix and don't just list a bunch of equations and axioms, I'm looking for a more intuitive understanding. Thanks :oldsmile:
     
  2. jcsd
  3. Oct 5, 2015 #2

    BvU

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    1) (Not too much at once: single - stepping is what's required here !)

    The Lorentz transform sets the wave in motion all right, but the probability density is still a constant. So we have a perfectly determined momentum and a completely unknown position, still in agreement with H.U.P. So no answer to your "Where is the particle" !

    The author then adds another wave, only differing in direction. Result: standing wave. As you can for example get when you enclose one "thing" in a box. Bounces back and forth between the walls. Probability of finding "it" at the nodes is zero, so not constant in space, but still you don't know where it is. Boundary condition: probability is zero at both walls -- in other words: the wavelength has to fit exactly.

    Let this sink in.

    Your question: what particle are we talking about?
    Can be anything. Teachers like to let students calculate allowed wavelengths for visible objects (marble, dust particle, even planets :smile: ). Thanks to the small value of the Planck constant you get outrageous results (fortunately).

    Good! Helps you to keep an open mind. Remember: we are not trying to understand what "it" is, just trying to understand how to describe "it" 's behavior. "Embodiment" of a particle doesn't have much meaning in this descriptive context, so I would lean towards the -- also questionable -- "pocket"...

    By the way, I really like the video. On the other hand, having grown up without such representations, I see a risk that such visualizations hamper the imagination and then the abstraction suffers.[/quote][/quote]
     
    Last edited: Oct 5, 2015
  4. Oct 5, 2015 #3

    BvU

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    On further reading, I suspect the video made you think the forward moving/backward moving wave components are necessary. Not so. Depends on the boundary conditions (this is wave mechanics!).

    I guess you are being introduced into quantum mechanics, so I advise to forget about the Dirac equation for a while. In fact, you want to erase the whole paragraph you named 2). The statement
    makes no sense.

    I mixed up 1) and 2) in my preceding reply, sorry. No embodiment, and no pocket either. Hope I didn't make things more complicated for you by introducing the particle in a box example ( it was in fact meant to make things somewhat easier to imagine... what with the left moving/right moving waves and such. And you do have a 3) about that).

    Will now proceed with going through 3)
     
  5. Oct 5, 2015 #4

    BvU

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    Too many things at the same time, again. Do we, really ? Usually the textbooks treat this as a first example of working out the wave equation for a particular potential function. In general in combination with separation of time dependent and time independent parts

    What the given potential energy function influences is the development of the wave function. Simply because it is present in the wave equation. Nowhere is anything affected that has to do with the identity of the particle. And a free particle is a free particle. No interaction means V = 0 and nothing interesting changes. Am I misinterpreting your identity concept ?

    It does NOT disintegrate at all. It just spreads out. A consequence of the initial conditions and the wave equation. The probability of finding the particle somewhere remains exactly the same (namely 1, the area under the "bell"). But the probability of finding the particle at the expected location does indeed decrease. That's what you get if the momentum isn't specified with zero uncertainty!

    (But you may notice I now also speak of 'particle' . I am not immune to mixing up wave behaviour and particle behaviour, just try to be aware that they have to be considered as living in different compartments...)​
     
  6. Oct 5, 2015 #5

    vanhees71

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    I haven't read all the thread, but I think one should warn against this picture of "relativistic quantum mechanics". There is no proper interpretation of a "wave function" in relativistic QT as there is in non-relativstic QT (Schrödinger equation), precisely the wave-mechanics formalism (aka "first quantization" in the position representation) cannot describe particle creation and annihilation, which occurs in the relativistic realm all the time. The only working description we have today for relativistic QT is relativistic quantum field theory.
     
  7. Oct 6, 2015 #6
    Thanks BvU, those are exactly the kind of posts I was hoping to get. I was afraid the thread was going to die on the vine after 3 days of no action. This thread was lonelier than Mark Watney stranded on Mars with no potatoes :frown: Thanks for being a first responder!

    Yeah, I guess my post was a bit ambitious. A "storm broke loose" in my brain and all my frustration with quantum uncertainty came flowing out at once. I'll make it point to ask questions in smaller bites from now on :biggrin:

    I've been self-studying physics for several years now. My knowledge and understanding is pretty broad but not very "deep," meaning I know a lot of the terminology and general ideas but not a great deal of the hard-core mathematical rigor. I'm very much a right-brain thinker that likes to visualize physical processes and the mathematics that describe them, which works pretty well for classical physics but obviously not so well for QM. There's so much "waviness" going around in QM I'm just trying to get a handle on what's what? You got the waviness of the photon, the EM field (all fields actually), the wave function (obviously), the wave packet, the probability density, etc., etc.

    For me, I'm not really sure what the relationship is between a probability "matter wave" describing a particle, say electron (e. g., it's position, momentum, and energy), a particles representative "wave packet," and a regular old electromagnetic wave that an accelerated electron is associated with. And if that's not enough, we can add the time component with the oscillatory behavior of the phasors to the mix. So now we got all these things waving around in space in time and it's hard for me to try to distinguish what is what. But I'm doing my best. Posts like yours go a long way to ground my thinking on these matters, so thanks again!
     
  8. Oct 6, 2015 #7

    vanhees71

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    Indeed, the main obstacle in understanding quantum theory is that the only "safe intuition" we have is the mathematical formalism, and thus it has to be trained. Don't get too much distracted from metaphysical ideas about interpretation in the beginning. Start with the minimal statistical interpretation. The key is the mathematical formalism (abstract Hilbert-space-opertor-algebra approach with symmetries and group theory) and its realization in various ways (wave mechanics=position representation, matrix mechanics, path-integral formalism, QFT). You can build some intuition via classical analogies, but you need also to keep in mind, where this classical intuition becomes inaccurate and even misleading. Also start with non-relativistic quantum theory first. Without a good understanding of it, there's no chance to get the relativistic case right, where you must do QFT. There's no conceptual clean way of a relativistic wave mechanics as in non-relativistic physics, except for low-energy approximations, where particle creation and destruction is suppressed due to a lack of energy. To understand electromagnetic radiation on the quantum level requires relativistic quantum field theory. Fortunately you come very far without it, doing atomic physics first, where the semiclassical approximation (classical em. waves and quantized electrons) is sufficient for a broad range of physics.
     
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