Reference Frames and the De Broglie Wavelength If the de Broglie wavelength of a massive particle, is h/mv then doesn't that mean interference patterns and everything change their properties depending on the velocity of the observer?
How can QM be provide consistent predictions in different reference frames with different observers expecting different wavelengths?
 A: This turns out to be a classic example of why we need to use complex numbers for the values of wavefunctions.
There are three basic facts that we have to start with:


*

*The wavefunction $\Psi$ is not observable. Only $|\Psi|^2$ is.

*The wavelength changes according to the rule $k\rightarrow k+(m/\hbar)v$, where the wave number $k$ is $2\pi/\lambda$. (This is just the de Broglie relation plus Galilean addition of velocities, notated in terms of $k$ rather than $\lambda$ for convenience.)

*Distances between observable points are invariant, because this is nonrelativistic quantum mechanics. (There is no length contraction.)
This doesn't work with real wavefunctions. If you have an oscillating real wavefunction, then it crosses zero at certain points. Zero squared gives zero probability density, so the locations of these zeroes is observable. The distances between these observable points is invariant, but this gives a contradiction with 2, because it makes the wavelength invariant.
So we actually have to do this using complex-valued wavefunctions. To keep the observable probabilities invariant, we have to have a transformation law for $\Psi$ that only changes its phase. We do this using the transformation law $\Psi\rightarrow e^{-ikx}\Psi$, which behaves as in 2.
In the example of double-slit interference, the fringes all stay the same, because the probabilities are invariant.
