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A very curious thing about quantum physics, 1920’s style, is that it can create observable interference patterns that are characteristic of overlapping waves. It’s especially curious because 1920’s quantum physics (“quantum mechanics”) is not a quantum theory of waves. Instead it is a quantum theory of particles — of objects with position and motion (even though one can’t precisely know the position and the motion simultaneously.)

(This is in contrast to quantum field theory of the 1950s, which [in its simplest forms] really is a quantum theory of waves. This distinction is one I’ve touched on, and we’ll go into more depth soon — but not today.)

In 1920s quantum physics, the only wave in sight is the wave function, which is useful in one method for describing the quantum physics of these particles. But the wave function exists outside of physical space, and instead exists in the abstract space of possibilities. So how do we get interference effects that are observable in physical space from waves in a weird, abstract space?

However it works, the apparent similarity between interference in 1920s quantum physics and the interference observed in water waves is misleading. Conceptually speaking, they are quite different. And appreciating this point is essential for comprehending quantum physics, including the famous double slit experiment (which I reviewed here.)

But I don’t want to address the double-slit experiment yet, because it is far more complicated than necessary. The complications obscure what it is really going on. We can make things much easier with a simpler experimental design, one that allows us to visualize all the details, and to explore why and how and where interference occurs and what its impacts are in the real world.

Once we’ve understood this simpler experiment fully, we’ll be able to discard all sorts of misleading and wrong statements about the double-slit experiment, and return to it with much clearer heads. A puzzle will still remain, but its true nature will be far more transparent without the distracting cloud of misguided clutter.

The Incoming Superposition Experiment

We’ve already discussed what can happen to a particle in a superposition of moving to the left or to the right, using a wave function like that in Fig. 1. The particle is outgoing from the center, with equal probability of going in one direction or the other. At each location, the square of the wave function’s absolute value (shown in black) tells us the probability of finding the particle at that location… so we are most likely to find it under one of the two peaks.

Figure 1: The wave function of a single particle in a superposition of moving outward from the center to the left or right. The wave function’s real and imaginary parts are shown in red and blue; its absolute-value squared in shown in black.

But now let’s turn this around; let’s look at a superposition in which the particle is incoming, with a wave function shown in Fig. 2. This is just the time-reversal of the wave function in Fig. 1. (We could create this superposition in a number of ways. I have described one of them previously — but let’s not worry today about how we got here, and keep our attention on what will happen when the two peaks in the wave function meet.)

Figure 2: The wave function of a single particle in a superposition of moving left or right toward the center. This is just Fig. 1 with time running in the opposite direction.

Important Caution! Despite what you may intuitively guess, the two peaks will not collide and interrupt each others’ motion. Objects that meet in physical space might collide, with significant impact on their motion — or they might pass by each other unscathed. But the peaks in Fig. 2 aren’t objects; the figure is a graph of a probability wave — a wave function — describing a single object. There’s no other object for our single object to collide with, and so it will move steadily and unencumbered at all times.

This is also clear when we use my standard technique of first viewing the system from a pre-quantum point of view, in which case the superposition translates into the two possibilities shown in Fig. 3: either the particle is moving to the right OR it is moving to the left. In neither possibility is there a second object to collide with, so no collision can take place.

Figure 3: In the pre-quantum version of the superposition in Fig. 2, the particle is initially to the left of center and moving to the right OR it is to the right of center and moving to the left.

The wave function for the particle, Ψ(x1), is a function of the particle’s possible position x1. It changes over time, and to find out how it behaves, we need to solve the famous Schrödinger equation. When we do so, we find Ψ(x1) evolves as depicted in Figs. 4a-4c, in which I’ve shown a close-up of the two peaks in Fig. 2 as they cross paths, using three different visualizations. These are the same three approaches to visualization shown in this post, each of which has its pros and cons; take your pick. [Note that there are no approximations in Fig. 4; it shows an exact solution to the Schrödinger equation.]

Figure 4a: A close-up look at the wave function of Fig. 2 as its two peaks approach, cross, and retreat. In black is the absolute-value-squared of the wave function; in red and blue are the wave function’s real and imaginary parts. Figure 4b: Same as Fig. 4a, with the curve showing the absolute value of the wave function, and with color representing the wave function’s argument [or “phase”]. Figure 4c: Same as Fig. 4a. The wave function’s absolute-value-squared is indicated in gray scale, with larger values corresponding to darker shading.

The wave function’s most remarkable features are seen at the “moment of crossing,” which is when our pre-quantum system has the particle reaching x=0 in both parts of the superposition (Fig. 5.)

Figure 5: The pre-quantum system at the moment of crossing, when the particle is at x=0 in both parts of the superposition.

At the exact moment of crossing, the wave function takes the form shown in Figs. 6a-c.

Figure 6a: Graph of the wave function Ψ(x1) at the crossing moment; in black is the absolute-value-squared of the wave function; in red and blue are the wave function’s real and imaginary parts. Figure 6b: Graph of the absolute value |Ψ(x1)| of the wave function at the crossing moment; the color represents the wave function’s argument [or “phase”].
Figure 6c: The absolute-value-squared of the wave function at the crossing moment, indicated in gray scale; larger values of |Ψ(x1)|2 are shown darker, with |Ψ(x1)|2=0 shown in white.

The wiggles in the wave function are a sign of interference. Something is interfering with something else. The pattern superficially resembles that of overlapping ripples in a pond, as in Fig. 7.

Figure 7: The overlap of two sets of ripples caused by an insect’s hind legs creates a visible interference pattern. Credit: Robert Cubitt.

If this pattern reminds you of the one seen in the double-slit experiment, that’s for a very good reason. What we have here is a simpler version of exactly the same effect (as briefly discussed here; we’ll return to this soon.)

These wiggles have a consequence. The quantity |Ψ(x1)|2, the absolute-value-squared of the wave function, tells us the probability of finding this one particle at this particular location x1 in the space of possibilities. (|Ψ(x1)|2 is represented as the black curve in Fig. 6a, as the square of the curve in Fig. 6b, and as the gray-scale value shown in Fig. 6c.) If |Ψ(x1)|2 is large at a particular value of x1, there is a substantial probability of measuring the particle to have position x1. Conversely, If |Ψ(x1)|2=0 at a particular value of x1, then we will not find the particle there.

[Note: I have repeated asserted this relationship between the wave function and the probable results of measurements, but we haven’t actually checked that it is true. Stay tuned; we will check it some weeks from now.]

So if we measure the particle’s position x1 at precisely the moment when the wave function looks like Fig. 5, we will never find it at the grid of points where the wave function is zero.

More generally, suppose we repeat this experiment many times in exactly the same way, setting up particle after particle in the initial superposition state of Fig. 2, measuring its position at the moment of crossing, and recording the result of the measurement. Then, since the particles are most probably found where |Ψ(x1)|2 is large and not where it is small, we will find the distribution of measured locations follows the interference pattern in Figs. 6a-6c, but only appearing one particle at a time, as in Fig. 8.

Figure 8: The experiment is repeated with particle after particle, with each particle’s position measured at the crossing moment. Each new measurement is shown as an orange dot; previous measurements are shown as blue dots. As more and more particles are observed, the interference pattern seen in Figs. 6a-6c gradually appears.

This gradual particle-by-particle appearance of an interference pattern is similar to what is seen in the double-slit experiment; it follows the same rules and has the same conceptual origin. But here everything is so simple that we can address basic questions. Most importantly, in this 1920’s quantum physics context, what is interfering with what, and where, and how?

• Is each particle interfering with itself?
  • Is it sometimes acting like a particle and sometimes acting like a wave?
  • Is it simultaneously a wave and a particle?
  • Is it something in between wave and particle?
• Is each particle interfering with other particles that came before it, and/or with others that will come after it?
• Is the wave function doing the interfering, as a result of the two parts of the superposition for particle 1 meeting in physical space?
• Or is it something else that’s going on?

Well, to approach these questions, let’s use our by now familiar trick of considering two particles rather than one. I’ll set up a scenario and pose a question for you to think about, and in a future post I’ll answer it and start addressing this set of questions.

Checking How Quantum Interference Works

Let’s put a system of two [distinguishable] particles into a superposition state that is roughly a doubling of the one we had before. The superposition again includes two parts. Rather than draw the wave function, I’ll draw the pre-quantum version (see Fig. 3 and compare to Fig. 2.) The pre-quantum version of the quantum system of interest looks like Fig. 9.

Figure 9: Two particles in a superposition of both particles moving right (starting from left of center) or both moving left (from right of center.) Their speeds are equal.

Roughly speaking, this is just a doubling of Fig. 3. In one part of the superposition, particles 1 and 2 are traveling to the right, while in the other they travel to the left. To keep things as simple as possible, let’s say

• all particles in all situations travel at the same speed; and
• if particles meet, they just pass through each other (much as photons or neutrinos would), so we don’t have to worry about collisions or any other interactions.

In this scenario, several interesting events happen in quick succession as the top particles move right and the bottom particles move left.

Event 1 (whose pre-quantum version is shown in Fig. 10a): at x=0, particle 1 arrives from the left in the top option and from the right in the bottom option.

Figure 10a: The pre-quantum system when event 1 occurs.

Events 2a and 2b: (whose pre-quantum versions is shown in Fig. 10b):

• at x=+1, particle 1 arrives from the left in the top option while particle 2 arrives from the right in the bottom option
• at x=-1, particle 2 arrives from the left in the top option while particle 1 arrives from the right in the bottom option

Figure 10b: The pre-quantum system when events 2a and 2b occur.

Event 3 (whose pre-quantum version is shown in Fig. 10c): at x=0, particle 2 arrives from the left in the top option and from the right in the bottom option.

Figure 10c: The pre-quantum system when event 3 occurs.

So now, here is The Big Question. In this full quantum version of this set-up, with the full quantum wave function in action, when will we see interference?

1. Will we see interference during events 1, 2a, 2b, and 3?
2. Will we see interference during events 1 and 3 only?
3. Will we see interference during events 2a and 2b only?
4. Will we see interference from the beginning of event 1 to the end of event 3?
5. Will we see interference during event 1 only?
6. Will we see no interference?
7. Will we see interference at some time other than events 1, 2a, 2b or 3?
8. Something else altogether?

And a bonus question: in any events where we see interference, where will the interference occur, and what roughly will it look like? (I.e. will it look like Fig. 6, where we had a simple interference pattern centered around x=0, or will it look somewhat different?)

What’s your vote? Make your educated guesses, silently or in the comments as you prefer. I’ll give you some time to think about it.

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