Quantum Mechanics & The Schrödinger Equation

If you watched Oppenheimer, you heard them talk a lot about the mysteries of quantum mechanics, and how crazy and counterintuitive it was when it was discovered. I thought pop science books only make it sound more mysterious. But at the heart of quantum mechanics is an equation called the Schrödinger equation, and it actually has a great intuition behind it that I'm going to try to describe!

What is the point behind quantum mechanics in the first place? It is really simple, and it is wild, to the point of unbelievable (actually to the point where plenty of people think we're still missing something fundamental in our interpretation of it, despite the fact that all its formulas work perfectly).

Here it is. Classically, we think of nature as "continuous": for example, a rainbow has all colors in it, it doesn't just "jump" from red to yellow. Bad analogy, but if you grow up with computers, you know they're digital and jump between 0 and 1. That's just not anybody's intuition of nature: nature looks and feels totally continuous. Computers are digital and not continuous, and thus they're just bad knock-offs of nature. Nature is analog.

But in the late 19th century, evidence piled up that at the fundamental level, nature is not continuous: at the atomic level, it becomes "discrete" - only certain values work. This is extremely counterintuitive and not at all what we encounter in day to day life.

Is it evidence that we live in a simulation? Who knows! Quantum mechanics might just be us running up against the "bits and bytes" of that simulation. (It's not, but fun thought.) But, it really is how our world works. A crazy early illustration is the Stern-Gerlach experiment from 1922 (when Oppenheimer was still a communist): particles (like electrons) have "spin", i.e., an intrinsic angular momentum (as if they were little spinning tops). Where there is a moving charge, there is a magnetic field, so this makes them like tiny, tiny magnets. Here is the thing: their magnetic field would have an orientation, sort of along their spinning axis. If you shoot electrons out of an electron cannon, you'd think that those "spinning tops" could be oriented any which way (= continuous!!). After all, you can turn a real magnet in any direction you want.

So Herr Stern and Herr Gerlach tried this experiment: they generated a bunch of wild, random particles that have spin (point 1 in the image below; in their case, they used silver atoms). They sent the particles through a magnetic field (point 3), and because of their intrinsic spin, the silver atoms got deflected based on their spin orientation. If spin could point anywhere, like it should intuitively, these particles should get deflected all over the place (point 4 in the image: a nice, smeared out mess on the screen). Instead, they arrived at exactly two locations (point 5 below).

Think about how bizarre that is. You don't do any sorting at the particle source here, these silver atoms just tumble out of the oven at 1. They fly all over the place, through an inhomogeneous magnetic field. And yet, they arrive at precisely two locations, and only ever there. The only explanation is: there are only two allowed directions for the spin to point. It can point up, or down. But absolutely nowhere else.

It is totally bizarre. At that level, the world is entirely, well, digital, at least when we measure. In fact, it is as if these particles' spin exists in its own "space", and that space has only two dimensions.

Particle spin is one of nature's "quantized" properties: it is discrete, not continuous.

Other evidence had piled up as well: Einstein's Nobel Prize was actually not for his theories of special and general relativity, but his explanation of the photoelectric effect, which showed a similar discreteness of nature.

So the early quantum mechanics faced a problem: the formulas they liked to use in physics were continuous, not discrete. But they needed formulas to describe these strange "discrete constraints" of nature. Equations like Newton's F = ma accept any kind of number, not just discrete values. You need some kind of construct that lets you use continuous math, but specify discrete values.

Waves let us do that! Here is a wave equation:

This is a one-dimensional complex wave equation. (Complex means it uses imaginary numbers: that just means it's really two independent numbers in one.) It has exactly two parameters you can adjust: the wave number k (the inverse of its wavelength), and the angular frequency ω (where its peaks and troughs are). The i is the imaginary unit of complex numbers, i = square root of -1.

This is how that wave looks (visualized by GPT-4 Code Interpreter!), at constant time t. It's really just a wave! (See how a complex number has a real and an imaginary part: it's just two independent axes.) But, we have exactly the "toggles" we want: we can give the wave a wave number k, and keep it at that. (If we increase k, the wave is curled more tightly; if we increase the angular frequency, a point that's sliding on this wave would be sliding faster.)

That is our "quantization": we now have a nifty tool that we can put one number into, and it outputs a wave that corresponds to that number.

So now we have to do something with those waves.

(By the way, if you're a software engineer and you hear "store and manipulate discrete values", you say "I'll use an array". Which is exactly what Werner Heisenberg, the guy who Oppenheimer outsmarted in building the bomb, first came up with, to describe quantum mechanics. An array is a matrix, and in those days, linear algebra wasn't really studied even in university, so Heisenberg wrote down complicated-looking systems of numbers, only for Max Born to eventually tell him "you're talking about a matrix, you should google it". Later, Heisenberg and Schrödinger realized they had described the same underlying math.)

But again, why use waves? In our intuition here, because they are constructs we can quantize - "machines" that have a toggle we can set to one number only (wave number k). But there is another reason: by the early 1920s, people knew that particles behave as waves sometimes. That came out of another wild experiment: the double slit experiment. In that experiment, you shine a light onto a screen with two slits. You get a pattern of peaks and valleys: light is a wave, so both slits become "originators" of one light wave each, and those waves overlap and interfere - meaning, the waves' own peaks and valleys overlap differently each place you get to. Same happens with water waves - no big deal.

But what is a big deal is that you can do that same experiment with electrons, and you get the same outcome. Which is profoundly strange: electrons seem like particles, so why wouldn't they just fly through the slits in straight lines, and that's it. No: they interfere, exactly like light waves do.

But it gets crazier: this even works if you just send one electron at a time. That is really completely bonkers, because what would a single electron even interfere with, if it's by itself? (What we know now is that even a single electron effectively takes every single path imaginable, and all of those paths interfere with each other, which creates the possible outcomes, and only when we measure, one of those will be chosen. You're welcome to think that the electron takes the other paths in a different universe, that is a serious interpretation of quantum mechanics.)

So this experiment gives us an important clue: modeling particles like waves actually makes physical sense, because the particles themselves seem to be telling us that.

We still are no closer to actually using waves to build ourselves a quantum mechanical equation. So let's do that. We start classical, i.e., continuous: a particle's energy that is moving with mass m and speed v through a potential V is

Let's ignore potential energy for now and use the particle momentum p = mv:

What do we know about the energy of a wave? From special relativity, a photon's energy is:

where p is the photon's momentum, and c the speed of light.

From Planck's study of blackbody radiation, he derived for the energy of a quanta of light:

where λ is the light's wavelength (basically its color!), and h the Planck constant.

(Hot tip: whenever you see c in an equation, it's special relativity physics, because that says that nothing can go faster than light, so you need the c in there. Whenever you see h, it's quantum physics. Believe it or not, early quantum mechanics was not relativistic, so c will not show up in the Schrödinger equation.)

So now let's boldly equalize them:

That's the k from the wave equation above: the wave number, or the inverse of the wavelength. So the momentum of the photon is p = hk. Meaning, a photon's momentum depends only on its wavelength.

Back to the energy of a particle, from above:

Is that the same p as in p = hk, the formula for the momentum of the photon? At first glance, no. In p = hk, we're talking about photons - which is light, not matter. In fact, photons famously have no "rest mass" (otherwise they couldn't fly at the speed of light; they don't exist if they aren't in motion). But in 1924, it was already clear that electrons behave just like light in the double-slit experiment, so they were behaving like waves. And Louis de Broglie had a momentous idea: let's treat them the same - every particle is also a wave! So: same formula for the electron's momentum p.

So now we have a formula for the energy of a moving particle based on its momentum, and another one based on its wavelength (whatever that is for a "thing" like an electron).

So far, this has all been physics. Now come the clever mathematical ideas. But first take stock of where we are:

• The Stern-Gerlach and other experiments show that for many sub-atomic properties, nature only allows "discrete", not continuous, values.
• A mathematical tool that allows you to specify discrete values is a wave: a single wave has a well-defined wavelength, for example.
• In fact, the double slit experiment with light showed us 200+ years ago that light is a wave.
• The double slit experiment with electrons shows that electrons also act like waves.
• So let's just treat electrons like waves, and use the mathematics of wave formulas to describe how electrons behave.

But now we have a math problem: take the formula we now have for an electron's momentum, p = hk (with k based on the "wavelength" of the electron wave). This now needs to become based on a wave that wiggles in time and space. Right now it's just a momentary, frozen snapshot.

Let's go back to the generic equation for a wave, from above:

This will appropriately wiggle in space x and time t. But, we know that an electron has a momentum, we can always just measure it. If the electron's momentum is p = hk, and now we want the electron to be a wiggly wave, then how would we measure k? We somehow need to "extract" k from that wave equation to calculate p. How do we do that mathematically?

We start using "operators": functions that we apply to other functions to extract what we need. Let's try this: we just assume that our wave equation Ψ(x,t) from above models the behavior of the electron. Then take the derivative to x of Ψ(x,t) in space x, and you get this (the strange symbol on the left is just the sign for "take the derivative"):

By rearranging:

See how the factor i times k came out in front of the wave function? We successfully "extracted" k from the wave function's exponential, by taking the derivative in x. Now multiply this with the factor (- i * h). Remember, i is the imaginary unit, and it's defined as i^2 = -1. So by multiplying again with i, we kill the i from this. And h is just the Planck constant from our momentum formula. When you do this multiplication, this becomes just the formula for p = hk! Here:

So: if you apply the "operator" -i * h ∂/∂x to a wave function Ψ(x,t), it will "extract" p.

How do you "extract" E? From Planck, we know E = hω. We extract that the same way - but now we have to take the derivative to time t (not space x) of Ψ(x,t)!

Here is our energy operator:

So: if you apply the "operator" i * h ∂/∂t to a wave function Ψ(x,t), it will "extract" E.

Now we have everything! Go back to our "classical" formula E = p^2 / 2m, for the kinetic energy of an electron. We are now going to say that we want wave functions everywhere. So we're going to replace the "classical" variables E and p with their "operator" equivalents. That is the magic of "quantization": 1) we know what the correct physical relationships are that we always want our physical values E and p to have to each other, but 2) we want the underlying variables to be waves. So we are constructing ourselves a formula that simply forces the physical values E and p to always be in the correct relationships to each other, but they can wiggle around as wave functions, and our operators extract the physical values from the wave functions. Start here:

Now replace E with its operator equivalent (left side), and p with its operator equivalent (right side), and have the operators operate on a wave function:

That is the Schrödinger equation!

What did we do here?

1. Classical mechanics tells us the energy of a particle from its momentum.
2. We have to "quantize", i.e., force that old formula to only accept discrete values.
3. Using waves makes physical sense because particles seem to behave like them. But waves also give us an elegant math tool to "quantize", because waves have a defined wave number k and angular frequency ω. Waves are continuous equations with discrete parameters.
4. Now we need to find a way to adapt our old particle energy formula to only admit waves as solutions. We do that by defining "operators": functions that, when we apply them to wave functions, extract the information we need.
5. We re-write our particle energy formula using operators, and a wave function that the operators act on. The formula forces the information we extract from the wave function into the relationships we require.
6. That is the Schrödinger equation.

Note what we did here: we didn't quantize (discretize) our underlying formula, we just forced it to only admit quantized solutions. In fact, that's how you use the Schrödinger equation: you take some scenario (like the hydrogen atom, with an electron flying around it), you figure out the scenario's boundary conditions and potential energy, you write those into the Schrödinger equation, and you can read off the wave function Ψ(x,t) that solves that particular variation of the Schrödinger equation. That wave function will tell you, at space x and time t, about the electron's properties. (Really, it means the electron is smeared out in space and time.) Or rather, the probability thereof, but that's a future story.

This is now, finally, why electrons can't just be anywhere around the atom's nucleus, but only in certain locations: because the wave that represents the electron has to be a standing, connected wave. That's how nature does it: if underlying everything there is a wave function that has to "fit", then that looks like "discrete" values when we measure it.