# Considering the wave nature of electrons, is it possible that doppler effect can apply on electricity too?

Just as light and sound in the form of waves can be explained with the doppler effect, I was thinking since electrons can also be thought of as waves, and since electricity is the flow of energy (or electrons, whatever), is it possible to have a have a doppler effect kind of explaination for electricity? Also, if electrons are considered a wave, how will you describe 'flow of electrons' ?

• The "doppler effect" for electrons would be an increase or decrease in the electron's measured momentum depending on your frame of reference's relative velocity to the electron. This changes the deBroglie wavelength of that electron. – enumaris Jul 24 '18 at 19:56

# Don't forget AC currents before you go all quantum

So electric current has two forms: one is a literal flow of electrons, and one is a changing electric field. If you look at a capacitor for example it is a break in a circuit: electrons cannot flow over it. But currents can, because the electric field in the middle changes, so that there is a continuous current through the capacitor.

Those currents are not only restricted to DC where current flows purely in one way; in fact the current coming out of your outlet is AC, where the electrons are constantly moving first left and then right and then left again. This is a classical wave and is subject to Doppler effects, which you could see if you had a way to "slide" your circuit along the power supply lines.

# But if you want to go quantum, Ehrenfest's theorem is great

Finally there is the actual wave nature of the electron. If you were looking at an old CRT television then inside there are these electrons being shot at a screen and each time they hit a color center they make that color light up. The electrons inside are generally nonrelativistic and might have wavelengths in the range of ~0.1 nanometers or so; we would say that they might have around 10 keV of energy, plus or minus half an order of magnitude.

You can also see those Doppler shifts but it turns out that something very interesting happens with nonrelativistic quantum mechanics. We have the Einstein relation that the energy of a particle is related to its quantum frequency by Planck's constant $h$ via $E = h f$, and de Broglie's relation that its momentum is related to its wavelength by the same constant, $p = h/\lambda.$ When one is thinking about the "speed" of a wave one generally forms the product $\bar c = f \lambda$ which is then $\bar c = E/p$, I am adding the bar to emphasize that for nonrelativistic particles this is generally much less than the speed of light, also often called $c$.

And then one can form the Doppler shift; if you are moving at a speed $u$ relative to some wave you see the frequency that wave with a characteristic shift $f = f_0 (1 + u/\bar c)$. Here we would say following the above Einstein law that also this means that the energy you see should follow $$E = E_0 \left[1 + \frac u{E_0/p_0}\right] = E_0 + p_0~u.$$

Now if you think about the underlying classical physics this is more or less exactly correct! So if you had a bunch of particles with masses $m_i$ moving with velocities $v_i$, and you start moving at a speed $u$ relative to them, you must form a different idea of all of their kinetic energies: before you thought that their energies were $$K_i = \frac12 m_i v_i^2$$ but now you think that their kinetic energies are $$K_i = \frac12 m_i (v_i + u)^2 = \frac12 m_i v_i^2 + m_i v_i u + \frac12 m_i u^2.$$ We could talk about the last term, $\frac12 m_i u^2,$ but it is not very important physically as in the energy picture we are mostly concerned with energy differences and not their absolute magnitudes. But the term in the middle is precisely the $p_i~u$ effect that we see above: the Doppler shift in the wave's energy is precisely the change in kinetic energy that we expect classically.

So this is a manifestation of Ehrenfest's theorem, which says roughly that quantum mechanics contains all of the central results of classical mechanics inside of it. When QM says that these particles have this strange wave frequency which is related to their energy, you can expect that those frequencies are subject to Doppler shifts -- but that those shifts replicate perfectly the changes in the classical energy picture that you were expecting to see anyway.