# Can de Broglie Waves have frequency, just because we know de Broglie wavelength formula? [duplicate]

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?

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 $$\frac{w}{k}=\frac{\hbar k}{2m}=\sqrt{\frac{E}{2m}}$$.