# electron wave and photon wave packet spreading

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?

Thanks

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.

• To me, this does not sound like a physical explanation. It's a mathematical one, which the OP already accepts. – garyp Feb 27 '17 at 22:19
• I agree, the math is clear how one gets the different dispersion relations, but the question remains why is it so? Can the difference be explained within the scope of quantum mechanics or does it need something like general theory of relativity or am I asking something that just can't be answered; like what is an electron? – Dubravko Babic Feb 28 '17 at 12:51
• What I wanted to show here is that mass seems to play a role in dispersion, and the reason is given by the fact that the formulation of energy in special relativity contains a mass term. If you want a more deep understanding you must look into QED, but I believe this simplification gets the essence of it. – Tool Feb 28 '17 at 15:12