Interference of overlapping wave functions

Question

I'm a physical layman trying to understand some of the consequences of quantum mechanics.

I understand that in the double-slit experiment, where we release individual photons in-phase, the probability to measure the arrival of a photon on a screen is zero on certain spots where the wave function becomes zero due to interference.

I reckon that "probability of arrival" really means "probability of interaction" with some electron, and that after such an interaction, the photon is gone and won't interact anywhere else.

Let's do another pair of experiments. First, with radio waves: We have two broadcasting antennas and in the middle of it a receiving antenna. The broadcasting antennas send the same sine wave in-sync so that the signal is perfectly cancelled out for the receiving antenna.

Theoretically, this should also be possible with a variation using laser light: Two laser shooting on a receptor precisely in the middle of the laser, so that the theoretical waves are cancelling each other out.

In practice, the receiving antenna won't receive anything. But that could have two interpretations:

1. The electrons in the receiving antenna actually don't interact with the ones in the broadcasting antennas at all - no radio photons are exchanged between them.
2. The electrons in the receiving antenna actually do interact with each of the broadcasting antennas as if the other one wasn't there, but the measurable net effect will be zero, ie. it's the electrons' current in the receiving antenna that cancel each other out.

Since radio photons don't come individually (in practice, I mean, as antenna create them), it's probably better to look at the laser variant of this experiment.

If the photons are ensured to come individually (we shoot them in an alternating fashion), it's obvious that the second thing will happen.

But if the photons are shot individually, but in-sync, it's not at all clear to me what one would see.

Answer 1

I have to say that it is not obvious that the second one happens if you have two "lasers" with the power adjusted so low that you get one photon at a time.

Now one problem is that I'm not sure about such a low power laser even existing, the design purpose of a laser is to send many photons in phase with each other, so if there is just one photon, I'm not sure it needs to be a laser. The point is that you want light from the left and from the right to come in in phase to interfere at the antenna, so you can take any low power source, even a microwave, and if it is coherent, send it through a beam splitter then arrange some mirrors so they come at your sample from opposite sides. You can put a detector in front of the beam splitter to verify that they come so slowly that they are coming through one at a time, but then when you remove the detector and let it go through the beam splitter I think you'll it interferes with the production at the source (and probably even breaks the coherence of the source). The real reason radio waves won't work is background, its hard to get interference because there are background waves. You can try to shield well (including low temperatures walls but radio waves will be hard to stop).

It's not going to do #2, even at low intensity (one photon at a time), because it is the total function that produces the probability, not a sum of probabilities of the "two" ways.

Answer 2

It's great that you're starting to learn about quantum mechanics. It's a truly fascinating subject.

It's important to first try and understand the classical theory of light and wave mechanics, at least qualitatively. This will help with your understanding of the quantum picture, since quantum mechanics deals a lot with waves.

Visible light, microwaves, and radio waves are all forms of electromagnetic radiation. The only difference among these waves is their frequency, and these categorizations are somewhat fluid and imprecise. So a "radio photon" doesn't really have much meaning in physics, and certainly doesn't differ from a "light photon" in any way other than its frequency. So the statement

radio photons don't come individually

isn't really correct. In a qualitative description of the double slit experiment, the frequency of the photon isn't very important.

With regards to the actual experiment, you're correct in saying that

the probability to measure the arrival of a photon on a screen is zero on certain spots where the wave function becomes zero due to interference.

But wave interference was a very well-understood phenomenon in classical optics for hundreds of years before the advent of quantum mechanics. So the mere fact that when you shine light onto the apparatus, an interference pattern appears on the screen, isn't and wasn't earth-shattering by any means - it's a trivial result.

However, using a low intensity laser beam, you can fire individual photons at the slits. This is where the physics gets interesting. When you do this you find that you still see the same interference pattern on the screen. However, if you detect which slit each photon goes through, the pattern disappears. This has no classical explanation, and is a beautiful illustration of quantum concepts.

To understand these results, you need some (basic) proficiency in wave mechanics and linear algebra. As this is a highly nonintuitive result (as are many results in QM), it's hard to come up with a satisfying qualitative explanation. I'll do my best to give a rough idea, glossing over some of the details.

Very roughly, the incident photon can travel through either one of the two slits. Its total wave function can be expressed as the wave function corresponding to the top slit + the wave function corresponding to the bottom slit. The interference pattern arises from interference between those two wave functions. A well-known consequence of measuring a particle state in QM is that if before the measurement the particle is in a superposition of states (state for top slit + state for bottom slit), the measurement will produce only one of the results (e.g. top slit), and after the measurement the particle's wave function is described by only that state. By detecting whether the photon went through the top slit, you are altering its wave function and destroying the superposition. Since there is only one wave function, there is no interference pattern.

Cool, right? Keep at it :)

Answer 3

Your question is confused in several ways. A laser does not fire photons. Rather, it produces a wave function that gives an amplitude $\alpha_j$ for any given number of photons $j$. The amplitude can be divided written as $\alpha_j=\sqrt{p_j}e^{i\phi_j}$, where $p_j$ is the probability of finding $j$ photons and $\phi_j$ is the phase of the field. In general, there is no single fact of the matter about where you will find the photon before you do the measurement: the system is actually described by the amplitudes.

If you lave two lasers running then the probability of a photon in any particular region is an integral of the sum of the wave functions in that location. If the wave functions cancel out in some region, the probability of finding a photon there will be low. Whether the wave functions cancel out or not is related to the phase of the field produced by the laser, not just its amplitude. If the amplitudes for the field from lasers 1,2 are $\alpha_{j1},\alpha_{j2}$, the total amplitude will be $\alpha_{j1}+\alpha_{j2}=\sqrt{p_{j1}}e^{i\phi_{j1}}+\sqrt{p_{j2}}e^{i\phi_{j2}}$, which depends on the phases not just the probability of each laser producing $j$ photons. (I am assuming here that the fields have the same polarisation etc.)

This brings up another point of confusion in what you said above. You describe the situation in terms of each laser firing a photon, but this is not accurate. Each laser produces some change in the electromagnetic field, but the field doesn't carry a label saying whether it is from laser 1 or laser 2. There could be some physical difference between the field from laser 1 and laser 2, e.g. - the polarisation is different. But in general there is no such thing as which laser a photon came from, there is just a sum of contributions to the field from different sources.

Now, you say that the lasers are firing the photons in sync, but this is not a well-defined idea. The only statement you can make that would match reality has to do with whether the wave functions are in phase or not, whether they have a constant phase relationship to one another: $\phi_{j1},\phi_{j2}$ could change over time so that they cancel out at a particular place at one time but not later.

There is dispute about what is happening in reality that corresponds to this description, if anything. I think the best way of understanding the issue is commonly called the many worlds interpretation of quantum mechanics. The whole of reality consists of a structure that looks a bit like a collection of parallel universes in some approximations. But the mere existence of those universes is not anything like a complete description, as illustrated by phases. See http://arxiv.org/abs/quant-ph/0104033 and "The Fabric of Reality" chapters 2,9,11 and "The Beginning of Infinity" chapter 11 by David Deutsch. Each contribution to the field carries some information about how it is related to other contributions: the phase.

Answer 4

I have seen your extensive discussion in the comments, especially about the use of the photon in the wave description, and that gives me rise to a response.

I understand that in the double-slit experiment, where we release individual photons in-phase, the probability to measure the arrival of a photon on a screen is zero on certain spots where the wave function becomes zero due to interference.

Single emitted photons also statistically produce the fringes on the screen. There is no question of interference.Moreover, photons with an energy content below pair formation do not interact or interfere. There is simply nothing to interfere. It is only so that the particles are deflected at the slit and hit in preferred regions of the screen.

Let's do another pair of experiments. First, with radio waves: We have two broadcasting antennas and in the middle of it a receiving antenna. The broadcasting antennas send the same sine wave in-sync so that the signal is perfectly cancelled out for the receiving antenna.

In fact, if you move the receiving antenna to the left or to the right, you get different intensities, from zero to twice the intensity of the transmitting antenna.

Theoretically, this should also be possible with a variation using laser light: Two laser shooting on a receptor precisely in the middle of the laser, so that the theoretical waves are cancelling each other out.

This happens only in the case that you modulate the laser light, analogous to how this happens with antennas

Recall how a transmitting antenna works. An alternating current pushes the surface electrons back and forth on the antenna and during their accelerations they emit photons. Every half period the electric field component of the photons is directed upward, the other half period downward.

In the receiving antenna, the signal doubles for the case when photons arrive from two emitters with the same photon direction. If the antenna is moved laterally, due to the runtime difference, a position can be found where at any time the incoming photons from two emitters always have the oppositely directed electric field and do not induce a current in the receiving antenna.

Briefly summarized

1. photons are oscillating quanta of energy whose electric and magnetic field components propagate in the form of waves.
   
2. EM radiation is mostly from unsynchronized emission of photons and it is impossible to measure an electric or magnetic field in such radiation.
3. EM waves originate from synchronously accelerated electrons and one can speak of interference from two sources.