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Why is deBroglie λ for electrons the same as λ for photons?

  1. Apr 28, 2017 #1
    Hi, I got the following question in my textbook: [translated]"Compare the wavelength of a photon and an electron where the photon and the electron have the same momentum".
    My thinking is the following:
    Firstly, pp (photon) = pe (electron).
    My textbook briefly mentions the extention of the mass-energy equivalence E2 = p2 c2 + m2 c4, so I go by this since the particles have different speeds. The de Broglie wavelength of the electron is given by fe = E/h = √(p2 c2 + m2 c4) / h. The wavelength of the photon is the same except that the mass is 0, so it reduces to fp = √(p2 c2) / h. Since m2 c4 ≥ 0 it follows that fe ≥ fp.

    My textbook says that the answer is that fe = fp tho. Their argument is that p = h/λ holds for both the electron and the photon. But they previously state that it only applies for massless particles. They derive it from E2 = p2 c2 + m2 c4 by setting m to 0. [translated]"[...]E = pc which holds for massless particles.[...] we find that p=h/λ".

    Why is my argument invalid and why does λ=h/p hold for the electron? Thanks!
     
  2. jcsd
  3. Apr 28, 2017 #2

    PeterDonis

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    Staff: Mentor

    No, it isn't. ##E/h## is a frequency, not a wavelength.

    Does it? With ##f_e## defined as ##E / h##? Or does it have a different formula for the de Broglie wavelength?

    And that is correct. Or, rearranging the formula, ##\lambda = p / h##. Not ##E / h##.
     
  4. Apr 28, 2017 #3
    The deBroglie wavelength as fe was a typo, I meant frequency. I now notice that I without really thinking about it assumed E=hf applied for the electron. I cannot this? (The book's definition of deBroglie wavelength is h/p, yes)
     
  5. Apr 28, 2017 #4

    PeterDonis

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    It does. What changes between the electron and the photon is the relationship between E and p; that should be obvious from the equations you wrote down in the OP. That in turn implies a change in the relationship between ##f## and ##\lambda##. (This relationship is often called a "dispersion relation" in the literature, and the difference I've just described is called a difference in the dispersion relation between massive and massless particles.)
     
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