Quantum Mechanics, Part 1 - Mechanics

Posted on August 6, 2022 by psu

I got it into my head that I should try to explain part of the problem with quantum mechanics on this web site. I am, of course, no expert on this subject at all. But I wanted to do a relatively simple and shallow (but mostly correct) treatment, like my category theory tutorial. So, over the last few months I’ve taken a few different shots at it but never found a way to wind it up into a single coherent train of thought. I wanted to thread my way through the physical puzzles to the mathematical formalism and then end up at the particular formula that, in my mind, sums up at least one of the problems.

I finally realized that trying to fit the whole thing into a single stream of words is beyond my talents as a writer, or at least not a structure that fits well into a single page on this web site. So I decided to split it up. So this first part is just about the move from “classical” mechanics to quantum problems … and then one or more future pages will be about the rest.

As with my other technical expositions on subjects that are not about computers, I am the furthest thing from an expert on this subject, I’m just organizing what I think are the most interesting ideas about what is going on here, and hoping that I’m not too wrong. I’ll provide a list of more better sources at the end.

Mechanics

To understand why quantum mechanics has puzzled people for so long we first have to go back to the mechanics that you might or might not have learned in high school or college physics. You remember …

F = ma,

all those stupid force diagrams with boxes and ramps and ropes and stuff.

It turns out that what all of this nonsense was hiding (which they tell you about sophomore year in college if you major in physics) is that every single one of these problems can be set up so you put some numbers into a single black box, turn a crank, and every answer that you ever needed falls out the other side. This magic box is a set of differential equations that describe how the system you have described evolves in time. I am not going to go into the details of how differential equations work, because honestly I don’t know them. But, for reference they look something like this:

\frac {d {x}}{dt} = \frac{\partial H}{\partial p}, \quad \frac {d {p}}{dt} = - \frac{\partial H}{\partial x}

Here, x represents position and p represents momentum (momentum is the mass of the object times its velocity … p = mv. For some reason this is a more convenient way to work than with the velocity directly). H is called the Hamiltonian, named after the mathematician who made it up: William Rowan Hamilton. It is a measure of the total energy in the system.

What the formula says, basically, is that if you have a thing and you can express the energy of the thing in the right way, then given any specification of an initial position and velocity of the thing, I can tell you exactly where the thing will be later and how fast it will be moving. All I need is a computer and the formula.

This basic set of mathematics is how we send probes millions of miles into space and have them hit a particular position over (say) Jupiter 5 years from now exactly when we think they will.

We will not really concern ourselves with the mathematical details of all of this, but there are two important thoughts to keep in mind here:

  1. In the above model, every “thing” that we study carries a definite value for these two attributes that we are calling “position” and “momentum.” These two values completely define the behavior of the objects that we are studying using this framework. So, the objects move through space on smooth and completely predictable paths, and it seems like their current state (position and momentum) is absolutely determined by their past state.

  2. More importantly, the model above directly computes all possible values of x and p that could possibly exist. That is, when you put your numbers in and turn the crank the numbers that come out are always, within the limitations of experimental error, the numbers that you see when you look at the real world. So you can, for example, throw a ball in the air and carefully track its position and speed at all times, and it will match the formulas pretty much perfectly. Not a lot of mystery.

By the end of the nineteenth century physics had developed two very successful models for how the world works: mechanics and electromagnetism and both fit into the mathematical and intellectual framework outlined above: behaviors determined by smooth and deterministic differential equations that compute values that are “real” in the actual world. Life was good.

The problem was that it didn’t work.

Quantum Mechanics

Quantum mechanics was originally born to describe the motion of atoms and things related to atoms. The development of the theory was driven by the experimental discovery of a host of behaviors that “classical” physics could not explain:

  1. The behavior of the so called “black body” radiation.

  2. The photoelectric effect.

  3. The puzzle of why atoms were stable, when according to classical E&M they should immediately collapse.

  4. The appearance of spectral lines that discrete frequencies in the spectrum of an atom.

  5. “Spin” and all that.

  6. The famous two-slit experiment.

And so on. All of these experiments are related to the “motion” of atomic (very very small) particles and radiation. The puzzling thing about these experiments with atoms and light was that while we think of atoms and their constituents as “particles” some of the behaviors that were observed only make sense if you model them as “waves”. On the other hand, classical E&M models light as a wave … but some of these experiments (the photoelectric effect) only made sense if light behaved more like a “particle”.

Over the first quarter of the 20th century various ad-hoc models and ideas were proposed to explain these things. But it wasn’t until the late 20s and early 30s that all of these ideas were codified into a more or less unified theory that we call quantum mechanics. The answer, it turned out, was to model material particles as waves, or “wave functions” that are solutions to a particular differential equation, the famous one from Schrödinger:

i \hbar \frac{\partial}{\partial t} | \psi(x, t) \rangle = H | \psi(x, t) \rangle .

Here the odd notation | \psi \rangle is used to denote the wave function of the quantum particle. I will go more into where this notation comes from in the next part.

The rest of the formula seems familiar enough on a surface level. H is again the Hamiltonian, and as before is related to the total energy of the system you are studying.

In fact, if you noodle around with this formula in just the right way you can come up with some mathematics that does a pretty good job computing the energy levels of the lines in the spectrum of the hydrogen atom. Recall that hydrogen is made up of a single proton with an electron whizzing around it. To explain the spectrum Bohr famously built a hypothetical model of the atom where electrons can only sit in a certain set of orbits that each have specific fixed energies. It turns out that when you set up the equations correctly you can find solutions to a version of the Schrödinger equation that give you wave functions at exactly these energies. You don’t get orbits, but you do get the so-called “stationary states” that are completely stable and match up with the spectral lines perfectly. So in some sense the electron is just sitting there waving around in some space in one of many possible fixed configurations. You’ve seen the pictures of the electron shells, right?

So what we have learned is that we can use Schrödinger’s equation and some smarts to tell us “where” the electron is in the atom. As ever, I will not go into the details. There are any number of books that will explain this to you. For example, Jim Baggot’s book is good for the physics point of view, while Stephanie Singer covers much the same material from a more mathematical viewpoint.

All of this makes you really want to believe that the wave function describes some sort of physical wave-like thing spread over all of space (x) and time (t) that will tell you something about the relationship between “where the particle is” and “what the energy is”. The fact that photons and even electrons create interference patterns that are very much like the ones you get from water waves in the two-slit experiment (see Feynman’s famous description here) makes you want to believe this even harder.

But, sadly, this is not so.

The Trouble With Quantum Mechanics

The waves in classical mechanics are an aggregate phenomena created by the motion of lots of things (air molecules, water molecules, etc) at once. Even more abstract entities like electromagnetic waves still have a sometimes visible macroscopic manifestation (let there be light!). In addition, as I mentioned before, the classical equations, in some sense, describe behavior that you can directly observe. You know the waves are moving through space on a particular trajectory because you can look at (say) the sky and see the light shining down on you.

The quantum wave function is nothing like this. Those complex numbers that are waving around are doing so in a space completely disconnected from the real world. In particular, they don’t tell you where the photon or electron is. Instead all they tell you is something about the chance that you have of seeing it somewhere if you look there.

But they don’t tell even you this probability directly. Instead, to get probabilities you have to compute something called the norm of the wave function, which is a measure of its overall magnitude … like its length if it were a piece of string. We write the norm of the wave function like this: |\psi| or |\psi(x,t)|. If you know how to compute it then the probability of finding an electron (say) at point point x in space would be

P(x,t) = |\psi(x,t)|^2 .

Computing this norm usually involves some kind of fancy integral. This interpretation of the wave function is called the Born Rule, and I’m not doing to go into the particular details of how one computes these things here. I will say though that this formula explains the interference patterns that you get in the two slit experiment. This computation turns up in a lot of “beginner” books on quantum mechanics, including the one by Feynman that I linked to above.

This rule feels like the luckiest in a series of lucky guesses. But it is undefeated in terms of experimental confirmation. Every experiment that has been done in quantum mechanics has amounted to thinking about a wave function, defining the right Hamiltonian, and then computing probabilities with the Born Rule, and the numbers are always right. Sometimes they are right to a ludicrous level of precision too.

In the famous double-slit experiment, for example, you send a beam of photons through one screen that has two very thin slits cut into it. Then you put a set of detectors a some distance away behind this screen. The mathematics of quantum mechanics will tell you that you should see an interference pattern on your detector array. It will even tell you the exact shape and configuration of the pattern. If you work hard enough you can probably compute this configuration with a stunning level of precision.

But quantum mechanics can’t really tell you anything about “what happens” to any single particle while it travels between the slits and the detector wall. The theory says nothing about it.

This, I think, is the first great mystery of the theory. It’s not so much that you can only compute and predict probabilities, there are many physical processes for which that is true. The real puzzle is that while the mathematics that I have hinted at above gets all the right answers, it does not appear to provide any insight into any actual physical process from which those answers can be derived. That is, your experiments always work, but it’s never really clear what is “really going on” in the “real” world.

Worse, as is well known, if you try and figure out what happens for yourself by (say) looking at each one of the slits to see which way the photon goes … the whole experiment falls apart and you get no quantum interference. Instead the act of measuring the position of the photon in some way seems to lock you into a history where all the photons suddenly take a single well defined path to the detector array, rather than creating the wavy interference that we got before.

This is, as you can imagine, a very unsatisfactory situation. Physics is supposed to tell you what happened and where things go. Classical mechanics seems to do this perfectly, right down to having an exact and satisfying connection between the mathematical model and what you observe in the real world. We get none of that in quantum mechanics. It is more like a computer program that always spits out the right answer but for which you do not have the source code, so you can’t reason about the exact mechanism by which the answer was generated.

In addition quantum mechanics appears to make you accept a world where the equations that tell you how systems evolve appear to be completely evolve one way (the smooth Schrödinger equation) when you leave them alone and another way (no interference) when you look at them. This is one aspect of the so called “measurement problem” and a lot of people smarter than me have thought about it and still find themselves confused. I am also mostly confused about this, but it will take a few more details to get at the core of why.

See you later, in part 2.

References

If what I have written makes no sense or you want to figure it out for yourself, here are some better sources than this humble web page.

  1. Travis Norsen’s Foundations of Quantum Mechanics is a great introduction to this material. A good combination of nuts and bolts physics and discussions of the conceptual issues.

  2. Baggot’s Quantum Cookbook is a good semi-historical treatment of early QM.

  3. Stephanie Singer’s algebraic treatment of the hydrogen atom is also enjoyable, but much more technical from a mathematical point of view.

  4. This series of lectures from Allan Adams at MIT is very good.

  5. Sean Carroll’s book is OK, as is Philip Ball’s book. They are both good non-technical explanations of the conceptual problems in the theory, to the extent that this is possible. Sabine Hossenfelder’s Youtube channel is also a good source for material at this level.

  6. On a more technical level, this paper about “Quantum Myths” is a nice antidote to the sort of woo woo mysticism that too much of the writing on this subject indulges in.

You should also read all the John Bell stuff, and various things by David Mermin.