Why is de Broglie wavelength related to momentum not energy? From relativity we have $E^2 = (cp)^2 + (mc^2)^2$ which for a photon ($m = 0$) becomes $E = cp$. From quantum mechanics we have for a photon $E = h\nu = hc/\lambda$. Thus together
$$
E = \frac{hc}{\lambda} = cp.
$$
If we want to generalise wave properties from photons to massive particles one assumes that the wavelength of a such particle is related to its momentum, but why not its energy? i.e. $\lambda = h/p$ instead of $\lambda = hc/E$? where $E$ is the total energy of the particle. So there seems to be two ways to generalise this.
There are two points about this:

*

*Massive particles will have wavelength even if at rest, so the wave vector $\vec k$ will have no preferred direction, but what about the particle being a point source of the wave in that case ?


*The order of magnitude, there will be a very large difference between the two generalisations, is it experiments that settled which one it is?
what is wrong with that?
 A: The wavelength $\lambda$ has a direction.
Or, more precisely, the wave number $$\vec{k}=2\pi
\begin{pmatrix}1/\lambda_x \\ 1/\lambda_y \\ 1/\lambda_z
\end{pmatrix}$$
is a vector quantity (with a direction perpendicular to the wave fronts).

(image from my answer to Significance of wave number?)
Therefore it makes more sense to relate the wave number $\vec{k}$ via
$$\vec{p}=\frac{h}{2\pi}\vec{k}$$
to the momentum $\vec{p}$ which is also a vector quantity,
and not to the energy $E$ which is a scalar quantity.
It also fits nicely into special relativity
where $(E/c, \vec{p})$ make up a 4-vector
and $(\omega/c, \vec{k})$ make up another 4-vector.
The four-vector $(\omega/c,\vec{k})$ is established by its 4-product
with the 4-vector $(ct,\vec{x})$
$$\phi=-\omega t+\vec{k}\cdot\vec{x}$$
giving the phase which is a 4-scalar.
But nevertheless, in 1924 when de Broglie brought up this hypothesis,
even this was still a speculative guess.
Only later it was confirmed experimentally that this relation
is actually true, not only for massless photons,
but also for massive particles.
The first experiment of this kind was the Davisson-Germer experiment
(1923-1927). It involved electrons (with known momentum $\vec{p}=m\vec{v}$)
scattered by the surface of a nickel crystal (with known atomic grid distance $d$).
From the observed diffraction pattern and the atomic grid distance $d$
they could calculate the wavelength $\lambda$ of the electrons,
and found it matched the wavelength predicted by de Broglie's $\lambda=h/p$.
A: Physics is not mathematics!
Perhaps somebody could contrive a mathematical theory in which the wavelength of quantum waves is related to energy, but,
Physics is an experimental science!
The de Broglie hypothesis matches what we see in numerous experiments, so we consider it a proven part of physics. Unlike mathematicians, we test our ideas against reality.
