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De Broglie and matter waves

  1. Apr 17, 2012 #1
    My book does not explain this and i cant rly find a simple explanation for this on google and my professor basically dodged my question:

    So the de Broglie wavelength formula basically says that matter creates waves with wavelength h/p, where p is the matter's momentum. However, when the velocity is zero, the denominator of the wavelength equation becomes infinite...by infinite wavelength im guessing this means that the wavelength is so long that it actually doesn't exist? So does that mean that a piece of matter only create a wave when it is moving? But that can't be right since motion is relative, so if something is not moving with respect to me, it is moving with respect to someone else, so that other person would be able to detect a matter wave but I wouldn't be able to...??
     
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  3. Apr 17, 2012 #2
    Aziza, when the particle has no velocity, it has no wavelength. Remember, the de Broglie wavelength refers to the probability density of the particle's wavefunction - essentially, the probability of finding it at a particular point along the wave. If a particle has no velocity, then you can be certain of it's position, and there is no wave.
     
  4. Apr 17, 2012 #3

    bhobba

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    Interesting way of looking at it - I wonder how it works out if you go to a FOR where it has a velocity - you go from knowing where it is to not knowing. IMHO you are mixing concepts from a theory that was a bridge to QM with the statistical interpretation of full QM. Indeed if you insist on that you run into the HUP which means it really cant be at rest anyway. IMHO what you are forgetting it is a relativistic property and even at rest a particle has energy and hence frequency - combined with the HUP I think it resolves the issue.

    Thanks
    Bill
     
    Last edited: Apr 18, 2012
  5. Apr 18, 2012 #4
    what is FOR and IMHO and HUP?

    Anyway thanks for the responses but I guess now my question has turned into, What EXACTLY is a "probability wave"? I mean I know that mathematically it describes the probability of "detecting" a particle in a certain volume, but what exactly is physically happening? Like for example my book says, "Suppose that a matter wave reaches a particle detector that is small; then the probability that a particle will be detected in a specified time interval is proportional to psi squared". But what exactly happens that allows for this "detection"? I mean, if particle has no definite position, then you shouldnt be able to ever detect it, because once it is detected, its obvious that at some certain point in time, that particle was exactly there, so the probability of the particle having been found somewhere else at that time was zero, since it wasnt there at that time!

    This idea that particle inherently does not have exact position is not sitting well with me. I mean I know that it would be hard to measure exact position, but it doesnt feel right that it doesnt actually have exact position. This probability wave idea seems like a good mathematical model, but does it literally describe physical reality? I mean I can perhaps see this if maybe space is also quantized, and then this would mean that we cant measure exact position of anything because the smallest thing that can exist is not actually an ideal infinitesimally small point, but fits into some smallest amount of volume possible, so we cant for example pinpoint exact location of a sphere. Idk, but that idea helps me remain sane while doing hw problems for this stuff lol. But do all serious/professional physicists actually interpret this probability wave/heisenberg uncertainty literally?
     
  6. Apr 18, 2012 #5
    Bill, even though de Broglie and Schrodinger developed their respective equations prior to the discovery of the rest of QM, this is the way that it has come to be interpreted (first by Heisenberg, Born, and Bohr in 1924-27). The wave represents the possible locations of the particle, which is why Heisenberg derived the Uncertainty Principle - the more momentum a particular particle has, the more 'spread out' the position is over said wave.

    Of course, the Uncertainty Principle forbids a particle to have zero momentum in the first place - this would, as I said in my original post, allow for the particle to have a definite position.
     
  7. Apr 18, 2012 #6

    Ken G

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    Actually, a particle with a definite momentum of zero does not have a known location.

    I would say that the deBroglie relation is really not more than a "signpost" to the formalism of quantum mechanics. What the deBroglie equation really does is connect a wavelength to a particle of definite momentum-- and if the momentum is definitely zero, then the wavelength is infinite. This doesn't mean it has no wavelength or no probability wave, it just means that the probability wave does not have any perceptible change with position. This also does not imply that you know where the particle is-- indeed, just the opposite. If the particle definitely has zero (or effectively zero) momentum, then you have no knowledge at all of its position. This is consistent with a probability wave that is the same everywhere-- no positions are picked out as any different from any other. Of course this is just an idealization-- in fact it is impossible to have a definite momentum, zero or otherwise, but it is a useful mental construct.

    Your question about "what is a probability wave" can be answered like this. In quantum mechanics, we speak in terms of an abstract "state" of a system, which embodies all the information we have about its history and/or preparation. This is also what we use to make predictions about measurements. When we want to make a prediction about a measurement, we first have to identify the possible outcomes of the measurement, and each state of the system that associates with each outcome. (If you want the jargon, the possible outcomes are "eigenvalues" and the states that associate with them are "eigenstates", of the "operator" associated with the measurement, but you don't really need to understand the formal meaning of these terms to get what I'm saying here.) Once you know each state that associates with a definite value of the measurement you have in mind, then the probability your initial state will give rise to that measured outcome is given by the square of the magnitude of a mathematical entity called the "inner product" of your initial state with the state associated with that measurement outcome. To get that "inner product", you have to refer to the information you have about the history/preparation of the system, for which the "state" is merely a shorthand label. So it all goes back to that history/preparation at some point.

    Now, oftentimes it is convenient to think in terms of a position measurement, and then the states of definite position are the ones we are taking the "inner product" of our initial state with. To get those inner products, we have to know how the information about the preparation/history of the state connects with position measurements. That information is conveyed in the "inner product" with the states of definition position, and those inner products are called the "wave function" or "probability wave" more loosely. So all those terms means is "the inner product between my initial state of interest and the states of definite outcome of a position measurement", and are physically interpreted as "the probability amplitude of finding the particle at x", or more precisely, "the probability amplitude of finding the particle within dx of x," since x is generally approximated as a continuous variable so there's no chance of finding the particle at exactly x but rather within dx of x.

    If you think it's strange to always refer to probabilities of finding the particle in some location, the truth is you often don't need to think about that, you can think in terms of probabilities of having a given momentum or energy, and there are plenty of situations where the "probability wave" is of no value at all and can serve to confuse you. What really matters is the "state", and its "inner products" with various states of definite measurement outcomes. However, since potential energy functions usually depend on x rather than on p or E, it turns out that the inner products with states of known x are often important to consider, hence the language "probability wave" or "wave function." States of definite momentum correspond to plane waves with the deBroglie wavelength when you take the inner product of the state of definite p with states of definite x, interpreted as the probability amplitude of finding a particle with known p at various x.
     
    Last edited: Apr 18, 2012
  8. Apr 18, 2012 #7

    Jano L.

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    Mark M,
    I think you misinterpret the uncertainty relation. If the wave function implies that a component of momentum along some direction is zero, it has to be constant along this direction. This means that based on such wave function, all positions along this direction are to be equally expected.

    Aziza, the wavelength is a relative quantity, such as the momentum. When we change the reference
    frame, the momentum and wavelength changes too. This is not so hard to understand, since the same thing happens already in wave optics - Doppler effect.

    When the object is at rest, the wavelength is not defined, but if it moves slowly, the de Broglie wavelength is very large.

    Jano
     
  9. Apr 18, 2012 #8
    Aziza,

    HUP = Heisenberg Uncertainty Principle
    IMHO = In My Honest Opinion
    FOR = Frame of Referance

    The process of a spread-out probability wave (called a 'wavefunction') reaching a more definite position, is known as wavefunction collapse. When the environment interacts with the wavefunction (for example, a detector bombarding it with particles) it undergoes decoherance, the driving force behind collapse. Remember, even after collapse, no definite position is attained - just one that is far more precise, but in response, you lose almost all certainty about its momentum.

    The second part of your question is one of interpretation. The ensemble position is one that does postulate that one should 'shut up and calculate' and ignore interpretation. Most interpretations describe the wavefunction as being real. For example, the leading, yet declining, Copenhagen Interpretation claims particle-wave duality - that the wavefunction is not really a particle or a wave, but both.

    Recent experiment such as SQUID suggest the wavefunction is real - macroscopic superpositions can exist.

    This is a very in depth topic, you should read this to get an idea:

    http://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics

    In response to the last part, I'd say the large majority of physicists that actively discuss interpretation belong to one that treats the wavefunction as real. (CI, Bohm, MWI, etc.)
     
  10. Apr 18, 2012 #9

    Ken G

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    The main issue that drives us to let go of the classical model that particles have exact positions is the "complementarity" of quantum mechanics. This means that it is very difficult to hold that a particle has an exact position and momentum at the same time, which is what classical mechanics generally imagines. So if you are "position-ocentric" you have a hard time being "momentum-ocentric" at the same time, and if you are "momentum-ocentric", you have a hard time being position-ocentric! Why would you want to hold that particles have definite positions but not definite momenta, or definite momenta but not definite positions? It seems more natural, once we open the door to indeterminacy, to open it all the way-- neither position nor momentum is exactly determined, both are partially determined in complementary ways. This stance is both useful and powerful in quantum mechanics, though other interpretations are always possible.

    What's more, I point out there is an even more important quantity than position or momentum, and that is energy. Energy is important because it controls the time-dependence of both classical (Hamiltonian) and quantum mechanics. But it is generally not possible for an interesting physical system to have a definite energy, in particular because systems with definite energy don't do anything interesting at all in quantum mechanics (they never exhibit changes of any kind in any observable property). So it turns out to be essential to allow indeterminacy in the energy, so why not momentum and position too? I think a big part of understanding the guts of quantum mechanics is becoming comfortable with the essential role played by indeterminacy. And why not? Determination comes from observation, and no observation is ever perfect, so we should have always expected our understanding of physics to be built on the back of indeterminacy. Classical determinacy was always a pretense-- measurement theory, chaotic phenomena, and thermodynamics all made that clear enough but it wasn't until quantum mechanics that we really needed to heed this lesson. IMHO.
     
  11. Apr 18, 2012 #10

    bhobba

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    FOR - Frame Of Reference - In this case another inertial frame (ie frame traveling at constant velocity) where the laws of physics are the same. My point was if it is at rest in one frame we can calculate where it is in an inertial frame in violation of the idea it has only a probability of being somewhere.

    HUP - Heisenberg Uncertainty Principle. If you know where a particle is then it says you know nothing of its momentum so it will scoot off elsewhere. This means you can't have a particle at rest and with zero momentum which also resolves your issue.

    IMHO - In My Humble Opinion.

    To answer your questions you need to study QM not the DeBrogle Theory. First why QM - check out:
    http://arxiv.org/pdf/quant-ph/0111068v1

    'Quantum theory, when stripped of all its incidental structure, is simply a new type of probability theory. Its predecessor, classical probability theory,is very intuitive. It can be developed almost by pure thought alone employing only some very basic intuitions about the nature of the physical world. This prompts the question of whether quantum theory could have been developed in a similar way. Put another way, could a nineteenth century physicist have developed quantum theory without any particular reference to experimental data?

    In a recent paper I have shown that the basic structure of quantum theory for finite and countably infinite dimensional Hilbert spaces follows from a set of five reasonable axioms. Four of these axioms are obviously consistent with both classical probability theory and with quantum theory. The remaining axiom states that there exists a continuous reversible transformation between any two pure states. This axiom rules out classical probability theory and gives us quantum theory. The key word in this axiom is the word “continuous”. If it is dropped then we get classical probability theory instead. The proof that quantum theory follows from these axioms, although involving simple mathematics, is rather lengthy.'

    Once you understand it is simply another form of probability theory then much of its weirdness dissipates. The wave particle duality does not exist - you only have particles. Waves are simply a theoretical device the equations sometimes throw up used in calculating the probabilities. A lot of research has been done into interpreting QM and a number of them are about - I personally hold to the Statistical Interpretation:
    http://www.kevinaylward.co.uk/qm/ballentine_ensemble_interpretation_1970.pdf

    Another excellent interpretation is Consistent Histories:
    http://quantum.phys.cmu.edu/CHS/histories.html

    It incorporates what is called Decohence which explains in a natural way our classical world. Most of Quantum weirdness is systems can be in states that are partly in say for example here and over there at the same time - this is the weirdness of Schrodinger's Cat. Decohence solves it by showing interaction with the environment, within a very short time, reduces such weird states to a simple probability statement - it is not in both places at the same time but rather is in one or the other with a certain probability. Specifically when a probability wave (caution here - remember it is simply a calculational device and does not actually exist - it is a representation of quantum state) reaches a detection device it interacts with the environment of the device which 'decoheres' the state (technically it is called tracing over the environment) and changes the state of the combined system (detection device and particle) to one where the particle has a definite position with a certain probability. You may have heard of the collapse of the wave-function issue - decoherence does not resolve that - but to an observer it looks like it does.

    Check out:
    http://ls.poly.edu/~jbain/philqm/philqmlectures/13.CH&Decoherence.pdf [Broken]

    Basically decoherence reduces QM to nothing more mysterious than flipping a coin and observing the result - at the classical level of our world you don't have this weird state of affairs where a coin is heads and tails up at the same time - it is one or the other with a certain probability.

    The following gives a lot more detail:
    http://motls.blogspot.com.au/2012/04/roland-omnes-and-qm-20-years-later.html

    Thanks
    Bill
     
    Last edited by a moderator: May 5, 2017
  12. Apr 18, 2012 #11

    bhobba

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    Just a few comments about Marks generally good answer.

    First the Copenhagen Interpretation does not consider the wave-function real - check out:
    http://motls.blogspot.com.au/2011/05/copenhagen-interpretation-of-quantum.html

    I thought so for a long time, but it was based on, as it turns out, incorrect accounts I had read.

    Secondly the Statistical and Ensemble Interpretation are the same thing.

    And finally its important to understand exactly what decoherence does and does not do. It does not give the particle a definite position with 100% certainty. Rather it changes the wierd state of affairs where the particle is partly in many different positions to one where it is in a definite position with a certain probability. This makes an observation no more mysterious than say flipping a coin where where know it will either be heads up or tails up - not a sort of wierd combination where both is partly true - but one where only one is true - but which one is true is given by probabilities.

    Basically QM has weird rules of logic where statements can be true and false at the same time (ie in one position and another simultaneously) but decoherence recovers the ordinarily sensible everyday world from it - admittedly probabilistic - but still intelligible.

    Thanks
    Bill
     
  13. Apr 19, 2012 #12

    Ken G

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    Decoherence doesn't resolve everything, however. Indeed, I would argue that decoherence does not resolve the fundamental questions that distinguish the various interpretations. Those questions are apparently not resolvable at all, at least at present, which is why we have so many interpretations that are all viable. One of the key questions that decoherence does not resolve is the unitary structure of quantum mechanics, when combined with the idea that even a system undergoing decoherence should be part of a larger system that is still closed, and so even though subsystems can undergo decoherence, supersystems should not (until they also become subsystems). In practice there really is no such thing as a closed supersystem, but that's not much of a resolution, because in practice none of the idealizations that physics imagines are actually true, physics must manipulate mathematical ideals in a way that provides practical results. The mathematical ideals of quantum mechanics get very hard to interpret when one includes the decohering agents into the system under study, particularly when one includes the most important decohering agent of all-- the observer themself. I think that we are nowhere close to understanding how the action of the observer affects, alters, and even defines, the physics that the observer will find successful, and that is why quantum mechanics remains difficult to interpret, even after understanding the crucial importance of decoherence.
     
  14. Apr 19, 2012 #13

    bhobba

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    It most definitely does not resolve everything. Most importantly it does not resolve the collapse of the wave-function issue in those interpretations in which it is a problem (it is not a problem in the Statistical Interpretation, Copenhagen, or Consistent Histories). It is the reason I still prefer the statistical interpretation even over my other favorite interpretation - Consistent Histories.

    What it does do however is make things intelligible at a classical level by removing off diagonal elements in the density matrix so an observation is reduced to a bog standard statistical observation with the usual rules of probability. It does not explain why it is probabilistic in the first place nor does it explain how it singles out the result it does. It simply removes this weird state of affairs where macroscopic objects like Schrodinger's Cat can be in this weird alive and dead state at the same time - it does not tell us why alive or dead was singled out.

    I do like to incorporate it into the Statistical Interpretation even though it does not require it because it does explain why the observation reduces to a standard statistical observation.

    Thanks
    Bill
     
  15. Apr 21, 2012 #14
    De Broglie started with an "internal frequency" in the rest frame which has the same phase everywhere (infinite phase velocity). When such a frequency is Lorentz-transformed into moving frames, relativity of simultaneity means that the phase in the new frame must vary spatially as well as temporally, ie it looks like a wave with finite wavelength. It's the same waveform in spacetime, but it's sliced at an angle instead of parallel to the wavefronts.

    The diagrams here are a good start for how to visualize the Lorentz-transformation:
    http://en.wikibooks.org/wiki/Specia...ation_and_length_contraction#De_Broglie_waves
    Reldebroglie.gif
     
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