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Sub-question if Yes: de Broglie wave is also often called the matter-wave. While we can find the frequency of an Electromagnetic Radiation from its wavelength $(c=\nu\lambda)$. $c$, in this case, is the speed of light in the vacuum which is a constant. Does a similar constant exist for matter that can be substituted? (I don't think so.)

Sub-question if No: Electromagnetic radiations display wave-duality nature. Are the wavelengths of their wave and de Broglie the same thing? Wouldn't that mean there exists frequency for wave?

John Rennie
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Adil Mohammed
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1 Answers1

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Let's consider a free particle (I would come to the case of a particle under a non-trivial potential later). The eigenspace of such a particle corresponding to a particular energy $E$ is two-fold degenerate, in particular, \begin{align} \psi^+_{E}&=e^{-i(Et-\sqrt{2mE}x)/\hbar}\\ \psi^-_{E}&=e^{-i(Et+\sqrt{2mE}x)/\hbar} \end{align} where symbols have their usual meanings. As you can see, the wave-function is indeed a wave-solution with the (angular) frequency $\omega=E/\hbar$ and wave-number $k=\sqrt{2mE}/\hbar$. Thus, we get that the rough analog of the speed of light would be either the group velocity of the wave-function or the phase velocity of the wave-function. As you can calculate, the group velocity is $\frac{dw}{dk}=\frac{\hbar k}{m}=\sqrt{\frac{2E}{m}}$ and the phase velocity is $\frac{w}{k}=\frac{\hbar k}{2m}=\sqrt{\frac{E}{2m}}$.

Two points of caution:

  • This framework cannot exactly handle the quantum mechanics of light because light is inherently relativistic and one needs to use quantum field theory to properly handle it.
  • Finally, in the case of a particle under a non-trivial potential, the solutions are not wave-like and thus, the de Broglie principle is not valid. See: Dispersion relation of QM in the presence of a potential.