Double-Slit Interference in Quantum Mechanics

[Robert_G]

Hi, there

Today, a friend came to me and asked the following questions and they made me confused.
If the source is strong, the light has to be considered as wave, and the double-slit interference (DSI) can be interpreted, and the conditions for the DSI is (1) same frequencies, (2) parallel polarization, and (3) constant phase differences.

If the source is week, one photon after one photon, for example. DSI will give us spots at the beginning; but give it a long time, the pattern will be formed.

Both of us never check it in the laboratory, we just read it from the book. However, In the "Quantum Mechanics" books, the above three conditions never be mentioned, As if those conditions does not mattered anymore.

In my opinion, All the three conditions are still needed. The character of frequency, polarization and phase difference are still with photons. Just look at the formulas of the creation and annihilation operators for the field. However, I can not* explain what the frequency, polarization and phase difference means to the particles (photons here), So I can not convince him, and my dear friend insists that those three conditions fade away in Quantum mechanics.

If we use the electrons in the DSI experiment, the coherent source is still need, right?

So, what you think?

*: The word "not" is added by author after the submission of this post.

 

[dEdt]

I would say that you certainly need all the photons (or electrons) to have the same frequency, but I don't think polarization or phase differences matter.

However, I can explain what the frequency, polarization and phase difference means to the particles (photons here), So I can not convince him, and my dear friend insists that those three conditions fade away in Quantum mechanics.

I assume you mean that you can't explain these terms.

In QM, particles are represented by a wave function ( see http://en.wikipedia.org/wiki/Wave_function ). The frequency and phase of a particle just refer to the frequency and phase of its associated wave function.

Polarization is a bit trickier. In particle physics, polarization is the same as spin, and spin is the intrinsic angular momentum of a particle.

 

[DrChinese]

You can see that those 3 conditions are not really necessary if you send through a photon at a time. So the question really is, to what extent are those conditions necessary if you send through a bunch of photons as a group. The answer is that you do not want the photons to destructively interfere with each other as this will mess up the interference pattern. Further, the frequency affects the width of the pattern, so mixing different of those would have an effect.

But all of these cases are still explained by QM. The only difference is that the "bunch at a time" case is also classically explainable.

 

[bhobba]

Those three conditions are not necessary for an explanation.

The fundamental laws of QM are all that's required:
http://arxiv.org/ftp/quant-ph/papers/0703/0703126.pdf

One of the issues here is the double slit experiment is often used to motivate the QM formalism, but the reverse isn't often done, ie going back and showing how that formalism explains it.

That's one reason I don't like the usual approach to QM that is sort of semi historical - you really need to unlearn the ideas that fathered QM such as matter waves etc etc - but that is rarely ever done. Much better to start with its conceptual core to begin with:
http://www.scottaaronson.com/democritus/lec9.html

Thanks
Bill

 

[Robert_G]

I would say that you certainly need all the photons (or electrons) to have the same frequency, but I don't think polarization or phase differences matter.




I assume you mean that you can't explain these terms.

In QM, particles are represented by a wave function ( see http://en.wikipedia.org/wiki/Wave_function ). The frequency and phase of a particle just refer to the frequency and phase of its associated wave function.

Polarization is a bit trickier. In particle physics, polarization is the same as spin, and spin is the intrinsic angular momentum of a particle.

I mean can't, sorry. better changing it.

 

[Robert_G]

I think the picture is more and more clear to me now, after reading all the above replies. The frequency of the wave function is the frequency of the light. Considering the Einstein's formula for the energy of photons and Schrodinger Equation, this just make sense. No matter whether wave function describes the probability of the single particle or an ensemble of them, the probability at a given position is oscillating at this frequency which coincide with that of corresponding classical light-wave. Since the path of the photon is limited by the "slit" which is not that narrow, the change of momentum is brought up according to the uncertainty principle, and the wave-function on the surface of the receiver is some kind of "wave-pack". and In that "wave-pack", the pattern appears. The frequencies should be same, and the coherence source is still needed. For the polarization, it seams to me that it is not matter in the quantum mechanics. Even if we can see those tiny spots on the receiver, they should be pattered spots which the chaos of the all kinds of polarization.