Electron wave and photon wave packet spreading

Question

I am looking for a physical interpretation of different behavior of electron and photon wave packets.

The dispersion relationship for a photon in free-space is linear ($\omega \propto k$), while for an electron (or any other massive particle) it is quadratic ($\omega \propto k^2$) (in free-space). If I form a (single) electron wave packet it will disperse in time (broaden with time of propagation), but a photon packet will not.

Apparently, any massive particle will behave the same way regardless of whether it has charge or whether it is a boson or a fermion. I would consider the dispersion relationship difference a purely mathematical explanation for this phenomenon, but is there a physical interpretation behind this?

Answer 1

A dispersion relation tells you the form of $\omega (k)$. Since $E = \hbar \omega$ and $P = \hbar k$ you can see it as a relation between the energy and the momentum.

Since we have from special relativity that $$ E^{2} = p^{2}c^{2} + m^{2}c^4$$

it is clear that we have $E = Pc$ for a photon. Also since the total energy of a free electron is $E=\gamma m c^2$ the kinetic energy is $E = (\gamma -1)mc^2$ wich reduces to $P^2/2m$ for $v<<c$

This way you can see how special relativity tells us that mass has a role in the dispersion relation, since rest (invariant) mass is the same for all observers in all reference frames. (It's the norm of the energy-momentum 4-vector in Minkowski space).

Returning to your question, you can see that photons follow the wave equation $$ \partial^2_t \Psi = v^2\nabla^2 \psi $$ whose solutions are transverse waves, wheras free electrons follow the Schödinger equation : $$ i\hbar\partial_t\Psi = -\hbar^2/2m\nabla^2\Psi $$ whose solutions are plane waves.

The dispersion relation is medium-dependent, for instance light is dispersionless in vacuum but not in matter, so in general $$v (n) = c/n$$ where $n$ is the medium's refractive index. For waves following Schrödinger's equation the dispersion relation is given in general by special relativity. This is why massive particles have a different dispersion than electromagnetic waves for example, and because massive particles have a phase velocity $v_\phi = \omega/k$ that depends upon the wavelength they broaden with time propagation.

(Edited a lot of times) I do not answer questions often, so I hope this is helpful.

Answer 2

I can give you two physical explanations.

One crispy:
Well, the photon would disperse in vacuum, it just does not have the time for it. The photon in vacuum travels at the (vacuum) speed of light. This means that whatever happens within the photon's world doesn't get to us, since its time is frozen. If the photon is smiling, you would see it always smiling. Therefore, the width of the photon remains frozen to you, for the whole photon's life, whatever the dynamics that would hold within the photon's world is.
On the other hand, in matter photons are slowed down. This entails that their time gets unfrozen. In turn it means that, whatever happens in the photon's world, leaks out to our world: as a consequence, we see the photon's wavepacket disperse in matter.

One more technical:
A wavepacket, be it related to massive or massless particle, is made of states with different momentum. Electrons (as well as any massive particle) disperses in vacuum since states with different momentum travel at different speed in vacuum: Some states are slower, some faster, resulting in an overall state getting wider. Photons, on the other hand, are simple objects in vacuum: Any photon travels at the same speed, irrespectively of its momentum. Therefore all components of the wavepacket travel at the same speed, thus keeping the wavepacket firm, its shape unchanged.
In matter, photons behave like massive particles, exactly because, in matter, photons with different momentum travel at different speed: photon's wavepacket disperses in matter.