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Frequency = mass or does it?

  1. Apr 6, 2014 #1
    Here's a puzzle that has been bothering me lately. I take as given the following:
      • [itex]E = h\nu[/itex] (Planck) energy equals frequency
      • [itex]E=mc^2[/itex] (Einstein) energy equals mass
      • [itex]mc^2 = h\nu[/itex] (de Broglie) mass equals frequency
    1. The phase frequency of an electron is altered by an electromagnetic potential. For a static electric potential [itex]V[/itex], this causes a phase shift [itex]\Delta\phi=-\frac{qVt}{\hbar}[/itex] that corresponds to an increase (decrease) of the electron phase frequency when [itex]V[/itex] is negative (positive). This is the same frequency change that shows up in different-energy solutions to the Schrodinger equation.

    Combining (2) and (1c), we seem to have the implication that an electrostatic potential should change the mass of an electron. We know the frequency is changed. We know frequency equals mass. We should be able to logically conclude that the mass is changed.

    Now clearly, that conclusion is wrong. If an electric potential changed the mass of electrons, then we would see changes in the size, mechanical strength, color, and other physical properties of the spheres of Van de Graaff generators when we charge them up, but this doesn't happen.

    Further, I know directly that the charge/mass ratio of electrons is not altered, because I ran that experiment myself in 2010, inside the sphere of the giant Van de Graaff at Boston Museum Of Science. I got a solid null result, with the only noticeable effect being a slight fuzzing of the electron beam (probably attributable to EM noise from the nearby generator). Within the 5% accuracy of the equipment, there was no observable change from about -1 MV to +1 MV potential. Since the mass of an electron is about half an MeV, the prediction from (2) is that at -1 MV the frequency should be roughly 3 times as great as for an electron at 0 V and therefore(?) by (1c) the mass should triple. If this effect actually existed, it should not be subtle and hard to detect.

    So, I'm not confused about the physical reality. I hope. What I am confused about, is why there is an apparently valid argument that the mass should change, when it doesn't. Why, in this case, does frequency NOT equal mass? Does the de Broglie relation not apply universally? If so, when does it apply, and when does it not?

    Am I just missing something really simple and obvious here?
  2. jcsd
  3. Apr 6, 2014 #2


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    1a. The Planck relation is for waves.

    1b. The Einstein equation given is for stationary objects with mass.

    1c. I don't recognize this is being from de Broglie; see http://en.wikipedia.org/wiki/Matter_wave

    In 2 you are using the electron as a wave; the Schroedinger equation can be "derived" or motivated by taking the Planck and de Broglie relations seriously - and then use them to transform a plane wave from wave vector & frequency to momentum & energy. Then to extract momentum and position from this "matter wave" the differential operators are obvious.

    Now write a generalized Hamiltonian for the system: everywhere that momentum appears swap in the momentum operator. This yields the LHS of the Shroedinger equation; with the energy operator you get the RHS.

    But there is a limited amount of information that can be obtained by simple algebraic substitutions.
  4. Apr 6, 2014 #3


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    How about considering the FULL relativistic equation?

    [tex] E^2 = (pc)^2 + (mc^2)^2[/tex]

    There is also a problem with simply plugging and chugging. Just because a quantity has the same symbol does not mean that it is the same thing. You should never simply do a substitution of variables without understanding the physical meaning of the equation.

  5. May 5, 2014 #4
    My 1c is equation 1.1.5 in Kraklauer's translation of de Broglie's PhD thesis (http://ebookbrowsee.net/de-broglie-kracklauer-pdf-d90199080); he says of it "This hypothesis is the basis of our theory". It's the starting point from which he derives everything else.

    Well, maybe, but the whole point of de Broglie's thesis was that just such a simple substitution gave the equation for matter waves and opened the door for wave mechanics.

    Perhaps my question could be rephrased as "What is the physical meaning of the change in phase frequency of an electron at potential?".
  6. May 5, 2014 #5


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    Since Dirac came up with his transformation theory in about 1927 that has been consigned to the dustbin of history - forget about it except as a historical curiosity.

    Read Ballentine - QM - A Modern Development for the correct modern take.

    No wonder - its a wrong theory - learn the correct one - not something that was the midfwife of the modern view.

  7. May 7, 2014 #6
    I don't recall anyone ever having claimed de Broglie's theory was wrong. Pauli pointed out some places were it was not consistent with Copenhagen practices and mindset. It was certainly discarded by the early QM cadre. It was used thereafter much more by chemists and other non QM specialists until Bell, Bohm and eventually de Broglie again opened the field once more.

    As ZapperZ has indicated, the full energy expression ##E=\gamma mc^2## implies that momentum can change which isn't incorporated into the mass. Does an electron moving towards a static electric potential imply a change of momentum or a changing phase velocity?

    Apart from that, it might not be correct to assume that ##mc^2=hf## is an identity, i.e. that a particle's mass is the same thing as the wave energy which can be measured at distances far away from where the particle is likely to be. Remember that de Broglie stated that the particle is accompanied by an energy wave, not necessarily that it is one. He gave rather convincing arguments that it cannot be.

    An analogy would be that ##F=ma## is not an identity. It implies a balance between force applied and change of momentum.
    Last edited: May 7, 2014
  8. May 7, 2014 #7


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    Well, the naive interpretation of the wave function, successful for non-relativistic quantum theory, fails for the relativistic theory. This is very easy to understand today, because we are used to the annihilation and creation of particles in collisions at relativistic energies. The naive interpretation of the wave function, used in non-relativistic quantum theory is for a situation, where you can deal with only one particle all the time. The most simple applications in QM1 are thus about one particle in an external potential. This doesn't work in relativistic quantum theory, because a pure single-particle theory must fail as soon as the energy of the particle you start with becomes large enough such that you can create particles. Then you must envoke a theory that can deal with such creation processes, i.e., with situations where the particle number is not conserved. The modern way is to use quantum field theory.

    According to quantum field theory you can even have spontaneous pair creation (i.e., the creation of electrons and positrons) by having a strong enough electric field. This Schwinger mechanism has, however, never been demonstrated experimentally yet. The reason is that you need very high electric field strengths for this effect, and those have not been reached in experiments yet.
  9. May 7, 2014 #8


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    It is wrong - not only for the reasons Vanhees elucidated - but right at its foundadtions. De Broglie hypothesized that any material particle has an associated wave with wavelength y= h/p. Electron diffraction etc supported this. But then, the electron at rest will have an infinite wavelength, and infinite wave phase velocity. This shows how the theory breaks down and needs to be modified. One must go to the full quantum theory. Beyond that are all sorts of other issues such as what Vanhees mentioned and what's the wavelength of electrons that are entangled with other electrons?

    One of the problems here is many textbooks give a historical take on QM proceeding from Plank to Einsten, to De Broglie, to Schrodinger, to Dirac etc. What they do not do is go back and see, in light of the now fully developed theory, the issues of its midwives and why they are WRONG.

    This sometimes shows up in a confusion about the double slit experiment where it is used to motivate QM but then do not go back and show how QM explains the double slit experiment. Because of that you see posts with people having this or that idea to explain the double slit experiment thinking they discovered some new theory. QM explains the double slit experiment and everything else besides.

    QM is what needs 'explaining' or rather a correct axiomatic development where its axioms are stated explicitly. And that's exactly what the reference to Ballentine I gave does - its developed from just 2 axioms.

    Just for the heck of it I will state them.

    1. Every observation is described by a resolution of the identity Ei in some complex vector space (these are so called Von Neumann observations - further developments of the theory leads to generalised observations where the positive operators Ei are not disjoint) or equivalently by a hermitian operator O, called an observable, such that its eigenvalues are the possible outcomes the observation. Interesting little exercise see why they are equivalent.

    2. The Born Rule. A positive operator of P unit trace exists such that the expected outcome of the observable O is E(O) = Trace(PO).

    To some extent the Born rule follows from axiom 1 via Gleason's theorem. Again an interesting exercise to see exactly what assumptions needs to be made to apply it.

    Last edited: May 7, 2014
  10. May 7, 2014 #9


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    It has no meaning.

    In terms of modern QM its simply a mathematical artefact of states represented as wave packets.

  11. May 8, 2014 #10
    I agree that Ballentine's presentation is a good characterization of the state of the art of core QM. But it does lack some material in how QM is used in related fields.

    I don't see a problem with infinite wavelength as the limit. If an electron were to come to rest then the energy associated with its movement would transition from being periodic in time-space geometry to being increasingly linear, even though that energy would dissipate entirely with the cessation of movement.

    Infinite phase velocity isn't at all problematic either really. We all know that it's a mathematical characterization of wave motion and not a physical movement of energy or information.
  12. May 8, 2014 #11


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    Its not a limit - its what it actually is - a wavelength of zero. You can always go to a frame where the electron is at rest and the wavelength makes no sense at all.

    This does not apply to QM because velocity is an operator- that's the rock bottom issue with De-Broglie's hypothesis - its a mishmash of classical and quantum.

    We have the issue of waves of what? QM - its simply waves of a theoretical device used to calculate probabilities so a constant function simply means the particle can be anywhere - for De-Broglie - your guess is as good as mine.

    I cant make any sense of that at all.

    Last edited: May 8, 2014
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