Is the Schrödinger Equation True?

If you squint, you can see the Schrödinger equation in a box on this page of Introduction to Quantum Mechanics by David Griffiths and Darrell Schroeter, which I “read” for my quantum experiment. I “explain” the equation here.

February 19, 2024. A while ago my girlfriend alerted me to a viral video in which a teenage girl named Gracie Cunningham announces, “I don’t think math is real.” Some of the math she’s learning in school, she suggests, has little to do with the world in which she lives. “I get addition, like, if I take two apples and add three it’s five. But how would you come up with the concept of algebra?”

While some geeks mocked Cunningham, others came to her defense, pointing out that she is raising profound questions about how mathematics relates to reality. Gracie’s complaints struck a chord in me because of my recent effort to learn quantum mechanics. Struggling to grasp eigenvectors, complex conjugates and other esoterica, I often wondered, as Cunningham put it, “Who came up with this concept?”

Reality, great sages have assured us, is essentially mathematical. Galileo declared that “the great book of nature is written in mathematics.” We’re part of nature, aren’t we? So why does mathematics, once we get past real numbers and basic arithmetic, feel so alien to most of us?

More to Gracie’s point, how real are the equations with which we represent nature? As real as or even more real than nature itself, as Plato insisted? Were the equations of quantum mechanics and general relativity waiting for us to discover them in the same way that gold, gravity and galaxies were waiting?

Physicists’ theories work. They predict the arc of planets and the flutter of electrons, and they have spawned smartphones and H-bombs. But scientists, and especially physicists, aren’t just seeking practical advances. They’re after Truth. They want to believe their theories are correct—exclusively correct—representations of nature. Physicists share this craving with religious folk, who need to believe their path to salvation is the One True Path.

But can you call a theory true if no one understands it? A century after inventing quantum mechanics, physicists still squabble over what, exactly, it tells us about reality. Consider the Schrödinger equation, which allows you to compute the “wave function” of an electron. The wave function, in turn, yields a “probability amplitude,” which, when squared, yields the likelihood that you’ll find the electron in a certain spot.

The wave function has embedded within it imaginary numbers, which consist of the square roots of negative numbers. You can’t find imaginary numbers on the line of negative and positive real numbers; nor do imaginary numbers map directly onto anything in the world. The wave function works, but no one knows why. The same can be said of the Schrödinger equation.

Maybe we should look at the Schrödinger equation not as a discovery but as an invention, an arbitrary, contingent, historical accident, as much so as the Greek and Arabic symbols with which we represent functions and numbers. After all, physicists arrived at the Schrödinger equation and other canonical quantum formulas only haltingly, after many false steps.

Moreover, the Schrödinger equation is far from all-powerful. Although it does a great job modeling a hydrogen atom, the Schrödinger equation can’t yield an exact description of a helium atom. Helium, which consists of a positively charged nucleus and two electrons, poses a three-body problem, which can be solved, if at all, only through extra mathematical sleights of hand.

And three-body problems are just a subset of the vastly larger set of N-body problems, which riddle classical as well as quantum physics. Physicists exalt the beauty and elegance of Newton’s law of gravitational attraction and of the Schrödinger equation. But the formulas match experimental data only with the help of hideously complex patches and approximations.

Maybe the best we can say of any mathematical theory is that it works in a particular context. That is the surprisingly subversive takeaway of Eugene Wigner’s famous 1960 essay “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.”

Wigner notes that Newton’s laws of motion, quantum mechanics and general relativity are extraordinarily, even unreasonably effective. Why do they work so well? No one knows, Wigner admits. But just because these mathematical models work, he emphasizes, does not mean they are “uniquely” true.

Wigner points out several problems with this assumption. First, theories of physics are limited in their scope; they apply only to specific, highly circumscribed aspects of nature, and they leave lots of stuff out. Second, quantum mechanics and general relativity, the foundational theories of modern physics, are mathematically incompatible.

Moreover, the “laws” of physics have little to say about biology, and especially about consciousness, the most baffling of all biological phenomena. When we understand life and consciousness better, Wigner suggests, inconsistencies might arise between biology and physics, implying that physics is incomplete or wrong.

My takeaway from Wigner’s 1960 essay, which remains as relevant as ever, is that physicists should not confuse their mathematical models with reality. That’s also the position of Scott Beaver, one of the commenters on Gracie Cunningham’s math video. “Here’s my simple answer about whether math is real: No,” says Beaver, a chemical engineer. “Math is just a way to describe patterns. Patterns are real, but not math. Nonetheless, math is really, really useful stuff!”

I like the pragmatism of Beaver’s view, which reflects, I’m guessing, his background in engineering. Compared to physicists, engineers are humble. When trying to solve a problem—such as building a new car or drone—engineers don’t ask whether a given solution is “true”; that terminology is a category error. Engineers ask whether the solution works, whether it solves the problem at hand more effectively than other possible solutions.

Mathematical models such as quantum mechanics and general relativity work extraordinarily well. But they aren’t real in the same sense that gold, gravity and galaxies are real, and we shouldn’t confer upon them the status of “truth” or “laws of nature.”

If physicists adopt this humble mindset, and resist their craving for certitude, they are more likely to seek and hence to find more more effective theories, perhaps ones that work even better than quantum mechanics. The catch is that they must abandon hope of finding a final formula, one that demystifies, once and for all, our weird, weird world.

Further Reading:

Conservation of Ignorance: A New Law of Nature

Is Ultimate Truth an Equation? Nah

The Delusion of Scientific Omniscience

Pluralism: Beyond the One and Only Truth

Theories of Consciousness, Gaza and My Cognitive Dissonance

My Quantum Experiment (free, online book)

Self-plagiarism Alert: This is an updated, vastly improved, free version of a paywalled column I wrote for Scientific American.