The Particle in a Box

From Wikipedia.

HOBOKEN, DECEMBER 2, 2024.  If I can understand the particle in a box, I can understand quantum mechanics, which means, maybe, I can understand, like, you know, life.

This notion popped into my head three years ago, and it’s still rattling around in there. Here’s the backstory:

Spring 2020, I set out to learn quantum mechanics with the math. I boned up on trigonometry, calculus, linear algebra and complex numbers. A complex number consists of a real number, which falls anywhere on the line of numbers running from negative to positive infinity, plus an imaginary number. An imaginary number is a multiple of i, the square root of -1.

You might have assumed that negative numbers don’t have square roots, because a negative number times a negative number is a positive number. But a while ago clever mathematicians said: Let’s just imagine there are numbers that, when multiplied by themselves, produce a negative number.

A complex number takes the form a + bi, where a and b are real numbers and i is the square root of negative one. Complex numbers inhabit their own version of a Cartesian grid, and they have lots of applications, including modeling quantum stuff. Complex numbers are key components of wave functions, mathematical widgets that track the swerves of electrons and other particles. When plotted over time, wave functions generate sinusoidal patterns, hence the name.

Got that? Okay. In the fall of 2020, I took a course at my school, Stevens Institute of Technology, called PEP553: Quantum Mechanics and Engineering Applications. The course text was Introduction to Quantum Mechanics, 3rd edition, by David Griffiths and Darrell Schroeter, a.k.a. Griffiths. This ain’t no pop-physics book, it’s crammed with equations and technical lingo (see below).

PEP553 crushed me, my math was too meager. But I still learned a few things with the help of our professor, Edward Whittaker. Typically, Ed would assign us a section of Griffiths, and then he’d go over it in class (via zoom, this was during the lock-down), asking us questions to make sure we got it.

Early on we learned about the “infinite square well,” commonly called the particle in a box. It’s what physicists call a toy model. Toy models are cartoonishly simple depictions of reality, but good ones can be revealing.

Griffiths puts it this way: “Despite its simplicity—or rather, precisely because of its simplicity—the infinite square well serves as a wonderfully accessible test case for all the fancy [quantum] machinery that comes later.”

Griffiths introduces the infinite square well.

Ed, our PEP553 professor, said the particle in a box serves as a terrific teaching tool; it has also inspired “quantum dots” and “quantum well lasers,” devices useful for sensing and communication.

The particle in a box shows how quantum mechanics diverges from classical mechanics--and from common sense. You place a generic, unspecified particle in a box with perfectly elastic walls, so the particle doesn’t lose energy when it bounces. The particle can have any mass, the box can be any size. Other than the particle, nothing is in the box; it’s a vacuum, free from gravity and other forces. Relativity doesn’t apply.

According to pre-quantum, classical mechanics, the particle can be anywhere in the box. It can move and bounce with any velocity and thus have any energy, kinetic or potential. It can sit still, with no energy at all.

Quantum mechanics, in contrast, says the particle must behave in certain constrained ways, dictated by its wave function. The particle can only inhabit specific energy levels, and it has a minimal energy that keeps it from sitting still.

Another key distinction: Unlike classical equations of motion, which show exactly where the particle is at any moment, the wave function yields only the probability that the particle will be in certain locations at certain times.

In its entry on the particle in a box, Wikipedia displays videos (see top of this page) that dramatize the gap between classical and quantum mechanics. In the classical-mechanics video, A, a red ball, the particle, bounces back and forth in the box along a straight line. Ho hum.

In the quantum videos, B-F, there is no ball; there are only undulating red and blue waves, which represent the imaginary and real components, respectively, of the particle’s wave function.

Confusingly, the video shows what we can never directly observe: the wave function. As soon as we look in the box, the wave function “collapses,” the red and blue waves vanish. We see only the particle, the red ball, in a certain spot. Before we look in the box, the ball, you might say, is imaginary.

The lowest possible energy level of a wave function is called its zero-point energy, which is never actually zero. Zero-point energy is the basis for wacky schemes, going back decades, for generating energy from nothing, that is, from a vacuum.

Zero-point energy also underpins inflation, a speculative theory of cosmic creation. According to inflation, our universe popped out of a wave function snaking through the primordial void. From nothing, comes something: 0 = 1. But who looked in the primordial box to bring our cosmos into existence?

Lately, I’ve been returning to that Wikipedia page to look at the video of the particle, or rather, wave function, in the box. Watching the red and blue waves wriggle, I have the eerie sense that I’m peering into my own head.

Some sages claim that quantum mechanics is the key to the mind-body problem, the mystery toward which all mysteries converge. That makes poetic sense. After all, the real and imaginary constantly collide in our minds, never more so than when we are in love or seeking love.

Is each of us a particle in our own individual box, or are we all in the same box? Is there, perhaps, only one box, and one particle, pretending to be many particles? Pondering these riddles, I feel my real and imaginary selves ricochetting around my skull, amplifying and interfering with each other.

For an instant, my wave function collapses, everything is still and clear, I know exactly who and what and where I am. I get it.

Then the wave function resurges, and once again I slosh back and forth, back and forth, within the walls of my box.

Further Reading:

The Ironic Interpretation of Quantum Mechanics

How Quantum Mechanics Helped Me Escape the Shitshow

Solipsism, Quantum Mechanics and Online Dating

Quantum Mechanics, Plato’s Cave and the Blind Piranha

On God, Quantum Mechanics and My Agnostic Schtick

Quantum Mechanics, the Chinese Room and the Limits of Understanding

Is the Schrödinger Equation True?

Conservation of Ignorance: A New Law of Nature

And when you’re done with those, check out My Quantum Experiment, from which this column is adapted.