# Probability amplitudes & light / particle wavelengths

• Nick.

#### Nick.

So this is basic question but the more I read the more I am confusing myself!

I was assuming that the wavelength of a photon was the same wavelength as the associated probability amplitude (although a complex number). So to make constructive interference it means one path takes say ten wavelengths more to arrive at the same point in the configuration space...but doesn't that mean path has taken more time than the other? As the photon travels at the speed of light I assume (again probably incorrectly) that if the time was measured it would be seen as taking the shortest path (I am postulating minimum action)...but if that is the case the other longer path hasn't even had enough time to arrive there so it can't interfere with it yet!

Sure - I probably have mix up a few things...some clarity and good further reading suggestions would great...

Thanks.

but doesn't that mean path has taken more time than the other?
Right. As a result, your coherence length has to be long enough to get interference. Your photon is not a point-like object.

I think your basic assumption of a photon as a particle that traverses a set path to get from point A to point B is incorrect.

Yes-So wouldn't the interference be in violation of the photon's fixed velocity? If it was interference from the same crest it would make some sense but to get the interference one path is longer than the other (with maximised probabilities at x+nλ).

And still - what time would be measured - the shortest path?

It may not travel a set path - hence the interference. But we can accurately measure the velocity of a photon - so that is a distance between two points (whatever path it takes in between). As the velocity is fixed, each path has different time...would the velocity equal the time to travel the shortest path?

So wouldn't the interference be in violation of the photon's fixed velocity?
No. Your assumption of a fixed position (or emission time) is wrong. Consider the classic equivalent to photons with long coherence length - a continuous wave emitted forever at the same frequency. It does not have a position at all.
each path has different time
Yes, but emission and absorption do not.

No I think we are getting sidetracked - I don't think I am going wrong with coherence. The simple fact that we get interference at all demonstrates that there are two (or more) different path lengths for the wave function. My confusion is about time - if the probability wave function path lengths relate to the actual light wavelengths then there is a difference in the speed of propagation. Although tiny in standard experiments (I calculate a difference of 2x10^-15 sec on a interferometer with 1m legs and 500nm laser) it still means something strange is afoot - either the wave function travels faster than light or the light is taking the longest path (unlikely).

if the probability wave function path lengths relate to the actual light wavelengths
It does not.
then there is a difference in the speed of propagation
No, and it would not even if the former was true.
(I calculate a difference of 2x10^-15 sec on a interferometer with 1m legs and 500nm laser)
What did you calculate, how?
either the wave function travels faster than light or the light is taking the longest path (unlikely).
Neither. Nothing strange happens.

Thanks mfb.

Can you expand your answers a little more so I can look them up - or perhaps you have a good reference? I am guessing that your inferring the wave is in a configuration space like a de broglie wave? (So no 'real' relationship with the speeds of particles - it becomes a calculation on probabilities).

Thanks

or perhaps you have a good reference?
A book about optics. This is so much easier to understand if you consider the electromagnetic wave as a (classical) field.

Thanks mfb...any good recommendations?

My favorite is good old Born and Wolf for classical optics and Scully and Zubairy and afterwards Mandel and Wolf for Quantum Optics.