Ch 7. Double Slit Enigma

The Double Slit Experiment probably needs no introduction for most of you, but I’m going to describe it in detail for those who aren’t familiar.  The experiment is known as being a kind of showcase for Quantum Mechanics, displaying its strange effects in a setting that’s easy to comprehend.  What seems to be missing is an explanation that’s also easy to comprehend.  For a long time, the “standard” view was “we can’t explain why this happens, but just go with it” (for example, check out Feynman’s presentation).  Now, the situation is much clearer. Due to progress in understanding entanglement and decoherence, we’re in a position to provide a model that explains the apparent weirdness of QM. Still, this is a very challenging topic, to say the least.  I’ve attempted to write this page with no mathematical notation, relying on visuals instead.  My hope is that at least some people will find it enlightening on some level.

The setup for the experiment looks something like what is shown in figure 7.1. Here, we see an “electron gun” (though we could use a gun that fired any type of “small” particle), a “screen” (with slits to allow particles through), and a “detector plate” (which has some way of determining where the electron hit). The entire setup is enclosed in an evacuated chamber, so that the particle doesn’t interact with anything other than what is shown.

Fig. 7.1 Setup for the double slit experiment.

Fig. 7.1 Setup for the double slit experiment.

 

As far as we are able to measure, electrons appear as “points”, with no extended length, width, or depth whatsoever. Also, there seems to be a limit to how repeatable the experiment is. If the gun was firing a large object like a baseball, it would be possible to have the object hit the detector in more or less exactly the same place every time (by making the gun very precise, eliminating noise, and so forth). When we use microscopic objects, there will be unavoidable randomness in where the detector registers a hit (assuming it makes it through the screen).

If we close one of the slits (say, the right side), and repeat the experiment many times, the detector will register hits in a distribution that looks something like figure 7.2. By itself, this is not very surprising, though we do see subtle indications of diffraction that foreshadow the bizarre effects to come.

When we open both slits, a very clear interference pattern emerges, as shown in figure 7.3. There are now regions where fewer particles arrive than in the case where only 1 slit was open! (If I had been more careful, I could have made it so that no particles appeared in the vertical striped areas).  Why does this happen? This is the 1rst major question that any interpretation of Quantum Mechanics needs to answer.

Figure 7.2

Figure 7.2. “Right Slit covered”

 

Figure 7.3. “Both Slits Open”

But there is another, perhaps even stranger issue: any attempt at learning which slit the particle went through destroys the interference pattern.  For example, we could add a second particle gun, firing a different type of particle, to our setup. We’ll locate it behind the screen and pointing to the left. We can also add a second detector plate that can register a hit from our new particle.  See Figure 7.4.  What we’ll do is fire the second particle from its gun with a speed and timing such that it will have a very high likelihood of hitting the particle from the first gun if that particle passes through the left slit, otherwise it will continue towards the second detector plate. In this way, we can use the 2nd detector plate to determine if the first particle went through the left slit. It turns out that the interference pattern becomes impossible to reproduce in this situation.

Figure 7.4. Rear view of our new experiment with 2nd particle

Figure 7.4. Rear view of our new experiment with 2nd particle

Now you might say, “Of course there’s no interference pattern now! Particle 2 knocks particle 1 out of the way, so this is really no different than the case where we covered up the right slit. There’s nothing new about this.” But here’s the crucial point: it doesn’t matter how much mass or momentum particle 2 has. For example, it could be a much lighter particle (or even a very low energy photon) that can’t substantially alter the path of particle 1. Even in this case, we will not see interference effects like figure 7.3, but will instead find a pattern that is a simple sum of what you would get from covering the slits up individually, like figure 7.5.

Figure 7.5. Both slit open, but with second particle added to the system

Figure 7.5. Both slit open, but with second particle added to the system

 

This is the type of pattern that “common sense” would predict you would see for the simple 1 particle double slit setup instead of figure 7.3. It is as though interaction with the second particle has caused the first particle to behave in a much more classical manner.  When we “look” at the particle, it behaves classically; otherwise Quantum effects manifest.  Why does this happen?

To summarize, there are 3 distinct mysteries to deal with:

  1.  Why is there some unavoidable randomness when dealing with “small” objects, that is absent when dealing with “large” objects?
  2.  Why does the interference pattern emerge when the second slit is opened? In particular, how is it possible that doing so actually restricts particles from arriving at certain areas that they had no trouble getting to with only 1 slit opened?
  3. Why does the interference pattern suddenly disappear when a second particle interacts with the first, even when the second particle doesn’t have enough momentum to substantially alter the path of the first particle?

Fortunately, there is a way of making sense of all of this, despite some very persistent rumors to the contrary.  What we are going to do is simple: set up the experiment mathematically (in accordance with the Schrödinger Equation) and solve it.  Then we’ll animate the solution and watch what happens.  As in the other chapters so far, our interpretation will be that the resulting “trajectories” are what is actually there.  As we will see, this will clear up all 3 mysteries.

To start with, can we produce the results of the “Single Slit Experiment”, figure 7.2?  For the animations below, I’ve set up a 1 particle, 2D environment with a barrier with a gap on one side.  The initial wavefunction is just a Gaussian wavepacket traveling to the right.  The first animation shows the probability density plot; the second shows the trajectories plot.

Unsurprisingly, the trajectories match the experiment perfectly.  We can also see the solution to the first mystery quite evidently in the spreading of the wavepacket.  The more narrow it is (or the less massive the particle is), the faster it spreads (see chapter 4 for more details about this).  This is going to translate to more randomness in where a trajectory will ultimately land on the detector plate, especially for very small particles such as electrons.

Next, we will simulate the double slit version.  Again, we have a probability plot followed by a trajectory plot.

 

Once again, the trajectories explain the perceived probabilities.  The interference pattern happens when the two wavepackets run into each other.  The collision causes the Quantum Potential to build up in the region where contact is made,  which will cause ripples in the density function, as noted in chapter 3.

That takes care of the second mystery.

The third one is trickier.  To simulate that situation completely, we would need a 4 dimensional wavefunction (2 particles x 2 dimensions).  The first problem would be how to plot either the probability density or the trajectories in 4D.  Another problem is the computer resources that would be needed.  The wavefunction from the last animation eats up about 500 GB of memory (including all time steps), and we’d need about a million times that to work in 4D, at similar energy levels for the system.  Fortunately, we can set up a similar situation in 1 dimension that has all of the features needed to understand what is going on.

For our new system, we’ll begin by letting particle A start out with 2 separated peaks and we’ll put particle 2 in-between those peaks (not worrying about how they got that way).  And unfortunately one other modification is going to be needed.  Ideally the second particle would have a mass that is tiny compared with the first particle.  The problem is that its wavepacket will spread out very quickly.  This means we would need to have the particles moving much faster to compensate (higher energies involved), which will necessitate a lot more computer memory being used.   Instead, we’re going to let the two “A” wavepackets move toward each other, and put the B wavepacket between them with zero (average) velocity.  This way we can let the 2 particles have the same mass, and we’ll still see what we need to see.  Also, to make the particle interaction very clear cut, we’re going to use a “billiard ball” style potential between particle A and B.  In other words, they don’t influence each other at all until they get close to each other, then the potential energy spikes.  Just to clarify, here are 2 animations showing what these 1D systems look like in isolation.  (Note that I’ve just plotted the 1D trajectories as dots right below the probability densities).

 

For the joint system, we create a wavefunction of 2 variables with initial state equal to the direct product of the initial states of the individual particles.  The image below shows what this looks like.  Note that I’ve projected the state against the “back wall” and the “left wall” of the bounding box of the plot.  Compare these with the 2 previous animations.

Figure 7.6. Initial state of particle A and B joint system

Figure 7.6. Initial state of particle A and B joint system

 

At last, we’re ready to animate the joint state.  The diagonal ridge represents our “billiard ball” potential.  It’s going to act like a wall that the wavepackets bounce off of.  This is because 1D billiard balls can’t pass through one another.

And the trajectories (this time the potential is just a diagonal gray stripe):

 

I imagine that I’ve lost some of my readers with the last two videos.  What do these plots mean, exactly?  What they’re really showing are sets of two particles.  Each black dot on the trajectories plot represents the positions of each particle in the 1D system.  The horizontal axis represents the position of particle A, and the vertical axis represents the position of particle B.  So the collection of dots are really a collection of systems of two particles.  Just to clarify, the plot below shows a sampling of some of these sets of two particles, moving around in their native 1 dimension (the red dots are particle A, the blue dots particle B).

 

In my opinion, this type of picture makes it easiest to imagine what a joint Quantum system looks like.  It’s a perfectly legitimate representation of the wavefunction, independent of any interpretation (though it does make the many worlds picture especially clear).

So, how do these plots help to clear up our third mystery?  In this 1D scenario, the third mystery corresponds to the fact that there is no interference pattern for particle A (the red dots in the last video).  The reason why is that the two wavepackets, as a whole, do not overlap.  There is plenty of overlap for the two groups of particle A, but by that time particle B has “split” into 2 distinct sets, with no overlap (see figure 7.7). In order for Quantum effects to appear, all of the system’s particles must be near each other (or distinct wavepackets need to overlap, to be more precise).

Figure 7.7. The two 'A' packets are overlapping and heading in opposite directions. However, the corresponding particle 'B' sets do not overlap. Hence the wavepackets for the joint system do not overlap.

Figure 7.7. The two ‘A’ packets are overlapping and heading in opposite directions. However, the corresponding particle ‘B’ sets do not overlap. Hence the wavepackets for the joint system do not overlap.

 

So, why does it matter that the B packets don’t overlap?  What is so special about packets overlapping?  It has to do with the way Potential Energy is distributed in the system.  There are 2 distinct types of Potential Energy, Classical and Quantum, which give rise to forces that shape the trajectories shown in our animations.  The Classical Potential behaves in the way you’d expect from ordinary physics.  Take another look at the last animation.  If you focus on one of the red-blue pairs, you’ll see they more or less just bounce off of each other when they get close enough (like billiard balls).  In that representation of the wavefunction, the Classical Potential only exists between particles lying on the same line, but it can propagate across large distances instantaneously (since we’re doing non-relativistic QM here).  The Quantum Potential is more subtle.  It is “wholistic”, in that sets of particles can influence other sets, but only if they are “nearby” in the sense that their configurations are close to each other.  The primary hallmark of the Quantum Potential is that it causes ripples in the density field.  The higher the magnitude of the QP, the shorter the wavelength of the ripples.  As with other, more familiar types of Energy, the QP can be converted to and from the other types.  In figure 7.8, we see the approximate moment when the QP was maximized during the particle collision.  There, we see what happens when a wavepacket collides with a wall, but the same thing occurs when 2 wavepackets collide; the Kinetic Energy for the system is converted to Quantum Potential Energy.  See chapter 3 for more details.

Figure 7.8. During the particle collision, the system's Kinetic Energy was turned into Quantum Potential Energy, which manifests as ripples in the density field. Note that diagonal lines represent constant distances between particle A and B. These ripples translate into some distances between A and B being encouraged, and others being not allowed at all. This is the reason for the famous "discrete" nature of QM.

Figure 7.8. During the particle collision, the system’s Kinetic Energy was turned into Quantum Potential Energy, which manifests as ripples in the density field. Note that diagonal lines represent constant distances between particle A and B. These ripples translate into some distances between A and B being encouraged, and others being not allowed at all. This is one manifestation of the famous “discrete” nature of QM.

 

Now, we can go back to our original 2D setup and understand what happened.  By making sure that particle B only interacted with particle A if particle A went through the left slit, we set things up such that the separation in ‘A’ caused a separation in ‘B’! This is because the ‘A’ that went through the left slit bumped particle ‘B’; the other ‘A’ did not.  If we had been able to compute the solution directly, we could have made something like our last animation, but with sets of 2 particles on planes instead of lines.  What we would see are 2 very distinct situations: either particle A would go left and bump B backwards, or A would go right and B would continue unaffected, to the left.  And this means the 4D wavepackets do not collide with each other.  Hence no Quantum Potential or interference pattern.

This process we’ve been describing, where superpositions of a particular system “leak out” into its environment is called “Decoherence”.  The subsystems involved in the process are said to become “entangled” as a result.

 

 

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