Dirac on Spreading Wave Packets

[anorlunda]

My layman's intuition tells me that wave packets normally spread out in space and disperse, except in special circumstances. Photons don't behave like that.

In The Principles of Quantum Mechanics, pp 124-125, Dirac discusses the equations of motions of a photon wave packet. He says:

For a given S, let us take a solution of (42) for which at some definite time, the density A^{2} vanishes everywhere outside a certain small region

Thus Dirac doesn't show that the packets don't disperse, he imposes it as a condition on the solution. I presume that the thing he left unsaid was "because that agrees with experiment," which is a compelling argument.

However, on the very next page he seems to waffle:

By a more accurate solution of the wave equation one can show that the accuracy with which the coordinates and momenta simultaneously have numerical values cannot remain permanently as favourable as the limit allowed by Heisenberg's principle of uncertianty, equation (56) of 24, but if it is initially so it will become less favourable, the wave packet undergoing a spreading. [see Kennary, Z.f. Physik, 44(1927), 344; Darwin, Proc. Roy. Soc. A. 117(1927)258]

"the wave packet undergoing a spreading" So what's the deal? Photon wave packets do or do not spread?

 

[Avodyne]

For a photon in free space, the wave packet does not spread. This is a consequence of the dispersion relation for a massless particle, $\omega = ck$, where $\omega$ is the angular frequency and $k$ is the magnitude of the wave number. This is also true in classical EM: wave packets propagate without changing shape.

See http://en.wikipedia.org/wiki/Dispersion_relation

 

[Jilang]

Sure they do, they spread out lots. Think of diffraction.

 

[The_Duck]

My layman's intuition tells me that wave packets normally spread out in space and disperse, except in special circumstances

Right.

Photons don't behave like that.

For photons, the associated wave packet represents something like the *probability* to find the photon at a given position. This wave packet of probability does spread out. But the photon is only ever detected at a single position.

Thus Dirac doesn't show that the packets don't disperse, he imposes it as a condition on the solution.

It sounds like Dirac is setting up initial conditions at t=0. He is not claiming that the wave packet doesn't disperse as you let time run forward. Indeed, in the next quote you give he discusses the fact that the wave packet spreads out as t increases.

 

[bhobba]

The Wikipedia article is actually quite good on this issue:
http://en.wikipedia.org/wiki/Wave_packet

Thanks
Bill