The Dice Are Loaded: Probability Waves

God does not play dice
~Albert Einstein

Einstein, stop telling God what to do!
~Niels Bohr

God playing dice
God playing dice (source).

This is part three of a multi-part series on quantum mechanics. In part one, I discussed how we discovered that light is both a wave and a particle. The dual nature of light suggests that massive particles like electrons might be waves too. In part two, I gave a theoretical underpinning to the dual nature of electrons: treating electrons as waves completes the Bohr Model of the atom and explains the Rydberg Formula. However, legendary physicist Richard Feynman once said:

It doesn’t matter how beautiful your theory is, it doesn’t matter how smart you are. If it doesn’t agree with experiment, it’s wrong.

The essence of science is to trust in the evidence. If we really want to believe that electrons are waves, we have to prove it. Fortunately, there is direct evidence that particles are waves. The experiment is electron diffraction, and it leads us to something far better. Through it, we learn what it means for something to be both a particle and a wave.

Two Slits and a Diffraction Pattern

Before we go on, I want to explore some properties of waves. I’ve previously discussed wave interference, but I’ll reiterate it here. If you remember my discussion from last time, feel free to skip the next paragraph.

Imagine waves as wiggles on a very stretchy string. If I try and push up on the string (make a wiggle that goes up) and you try and push down on the string (make a wiggle that goes down) at the same time, neither of us ends up moving the string as much as we intended. This is called destructive interference. Similarly, if I push up on the string and you push up on the string, we’ll probably stretch it quite a lot. This is called constructive interference. The process of overlaying one wave over another is called superposition.

Interference between two waves on a string
If you and I both try and make a wave on a string, we may interfere with each other. If, as in the image on the left, we both try and make the same wave, we may get a bigger wave than we intended. This is called constructive interference. If, on the other wand, we try to make waves exactly offset from each other, we may completely negate each others’ efforts. This is called destructive interference (source).

Light waves (and as we shall see, matter waves) behave this way too. If two light waves waves overlap, they can constructively or destructively interfere. I am glossing over a detail called polarization, but let’s ignore that for now.

Another wave property that we need to discuss is that waves are a little forgetful: if a wave is caused to pass through some sort of barrier, or move from one environment to another, the wave won’t retain its old shape, but will instead take on a new shape based on the transition. (I’m glossing over a lot of details here. For the experts, I’m talking about the boundary conditions of the wave equation.) When light is produced by a a single point, such as a light bulb, the light waves spread outward in every direction. The source is called a point source, and the wave is called a spherical wave. However, using a parabolic mirror or a lens, we can make some of that light form a column.

Lens columnates light
We can use a lens to take a spherical wave and make it a column (or collimate it).

We can also reverse this process, and take a collimated beam of light and make it look like it came from a point source. To obtain a perfect symmetry, we could do this with lenses or mirrors. However, we can also do it with a humble plate of metal (depending on the wavelength of the light). Imagine we have a collimated beam of light and it hits a barrier through which it cannot pass. If there is a small hole in the barrier (where small depends on the wavelength of the light), some light will pass through this hole and–here’s where it gets cool–the light will treat the hole as a point source. It forgets the shape it had before passing through the small hole. This is called single-slit diffraction, and in this way we can transform a collimated beam of light into a series of spherical waves.

Single-Slit Diffraction
In the left image, collimated light approaches a barrier from the left. When the light hits the barrier, some of the light passes through the small hole and the rest is blocked. The light that passes through the hole treats the hole as a point source, and propagates away from the hole in all directions as a spherical wave. Compare to the right image, where a light bulb generates a spherical wave.


Single-Slit Diffraction Animation
An animation of a collimated beam of light becoming a spherical wave. The wave fronts are displayed instead of light rays (source).

Now, things get really interesting when we add a second slit to the equation. In this case, the light forms two spherical waves–one for each slit–and the two spherical waves overlap. As we discussed above, overlapping waves interfere with each other, as shown below.

A Two-Slit Diffraction Experiment
A two-slit diffraction experiment. Columnated waves approach from the left, and after passing through the two slits, they form a pair of spherical waves. The waves overlap and interfere with each other. Bright blue is more intense light (constructive interference). Dim blue is less intense light (destructive interference) (source).

Because the peaks and troughs of the two spherical waves align just so (thanks to them coming from the same collimated beam), they alternate between constructive interference and destructive interference, as you rotate around the point between the two slits. The result is that, if you project the resulting two waves onto a screen, you see alternating bright spots and dark spots. The bright spots were generated by constructive interference, the dark spots by destructive interference.

The diffraction pattern from a two-slit experiment
The diffraction pattern from a two-slit experiment. The bright spots are places where the two spherical waves constructively interfered with each other. The dark spots are where they destructively interfered with each other (source).

This is, of course, all classical physics. What we’re really interested in is new physics: matter waves.

The Quantum Double-Slit Experiment

In 1961, Clauss Jönsson performed the same two-slit experiment, but instead of firing a beam of light at the slits, Jönsson fired a beam of electrons. By 1961, quantum theory was already fairly mature and we knew electrons had a dual particle-wave nature. However, because Jönsson’s experiment is by far the cleanest and most beautiful, I’m going to talk about it instead of one of many other experiments.

Jönsson found that, if he fired a single electron at the slits, it passed through (which a classical particle can’t do! The electron was fired at the center of the slits!) and hit his screen at a random place, but never in one of the spots that would be dark if Jönsson had performed his experiment with light (of the same wavelength as the electron). As more electrons passed through the two slits, they accumulated in exactly the same pattern as the interference pattern Jönsson would have gotten with light.

What’s going on? We know electrons to be particles—Robert Millikan and Harvey Fletcher even measured the electric charge associated with a single electron. And yet, Jönsson found that electrons in aggregate very much do behave like waves. And yet, Jönsson himself measured where the electrons hit by treating them as particles–he measured each electron as it hit the screen. We seem to have something new here. As Richard Feynman said,

“Electrons act like waves”–No they don’t exactly.
“They act like particles”–No they don’t exactly.

Throwing Dice

Scientists were stumped by the dual nature of both massive particles (like electrons) and light for a long time. Eventually they agreed on the ideas championed by Niels Bohr and Werner Heisenberg: what we now call the Copenhagen Interpretation. The idea is this: the wave nature of a particle is a probability wave. It represents the probability of an observer measuring a particle at a given place and time. Until the particle is measured, it is in every place it’s allowed to be. Once it is measured, the probability wave of the particle collapses and forces the particle into a single specific place.

The greater the amplitude the wave of a particle has, the more likely it is to be observed. You can think of this as rolling loaded die. The numbers on the die represent places the particle could be in. The die is weighted such that the places the particle is most likely to be come up more often than other places. When we observe a particle, we roll the die.

In an effort to disprove quantum theory, Albert Einstein famously quipped “God does not play dice.” But Einstein is wrong. Not only does God play dice, the dice are loaded.

A Mystery

This description of the world of particles is very strange–it is very hard to wrap your head around it–but there is overwhelming evidence that it is true. The probability of a particle being observed in a specific place is given by its wave nature, and the world of particles is inherently probabilistic. However, the Copenhagen interpretation raises a troubling question. What counts as an observer? In his experiment, did Jönsson collapse the probability wave when his electrons hit his screen? Or when he looked at the screen? We don’t know. Most physicists have decided this is a question for philosophy. Perhaps someday, though, we’ll be able to design an experiment to find out the truth.

Further Reading

If you’re interested in learning more, I have a couple of suggestions.


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