As a non-physicist, QM only clicked for me when I learned the linear algebra of one- and two-qubit systems and observables on those systems [1]. The math is accessible at a pre-calculus level, and the most "counter-intuitive" aspects of QM (interference, the uncertainty principle, entanglement) make perfect sense in this framework. I think it's the best starting point for understanding what QM really says.
[0] That's not to say they aren't important historically or for a complete understanding of QM. I'm talking about conceptual understanding for someone, especially an amateur, encountering QM for the first time.
[1] Chapter 2 of Nielsen and Chuang is what did it for me. It's heavygoing if you've never seen linear algebra before, but I don't know of a clearer explanation of the postulates of quantum mechanics.
Before quantum computers, the only reason to learn quantum mechanics was so that you could learn to explain the electron orbitals of the hydrogen atom. And after that proceed to understand how other atoms and other matter works.
And for this you need vector calculus, partial differential equations, Schrödinger equation, orthogonal functions (spherical harmonics, Laguerre polynomials) (with hint of the theory of Hilbert spaces).
The idea that someone would like to learn the basic principles of QM without applying it to solve the 3-dimensional probability distribution of electrons around the atomic nucleus, is rather new.
Scott Aaronson’s “Quantum Computing Since Democritus” makes a similar assertion about learning QM from a linear algebra perspective and completely sidestepping all the thorny interpretation questions. It’s mostly a book about complexity theory and what quantum computers actually buy you, but it gives a lot of insight into the mathematics of it.
That applied for almost every module, not just quantum.
My books went from 2 slit experiments, to particle in a box (1d and then 2d harmonic oscillators) with tunneling, and then the Hydrogen atom potential with electron orbitals as an extension, to perturbations for He, to Hartree-Fock and Born-Oppenheimer for large atoms and molecular models. Learning the <bra/ket> formalism can be useful for thinking about it, but it has to be motivated by the math.
I guess black body and scattering were pretty early too as motivation for the developments (as they historically were). Whereas EPR/Bell, Decoherence, and "Interpretations" were mostly discussed as interesting flavor and philosophical arguments.
The classic double-slit experiment is kinda bizarre when one considers the “single electron” case.
The modern record for a matter-wave superposition stands at a nanoparticle composed of roughly 7,000 sodium atoms (about 8 nanometers wide), achieving a massive "Schrödinger's cat" state at the scale of a small virus.
So then particles don't exist, and viruses also don't exist? Is "reality" just a dynamic pattern that doesn't exist in the naive sense? As astute stoics we should not worry ourselves with such matters, "shut up and calculate"?
There is something about quantum theory that is different in character from any physical theory posed before. To put a finger on it, the issue is this: The basic statement of the theory—the one we have all learned from our textbooks—seems to rely on terms our intuitions balk at as having any place in a fundamental description of reality. The notions of “observer” and “measurement” are taken as primitive, the very starting point of the theory. This is an unsettling situation! Shouldn’t physics be talking about what is before it starts talking about what will be seen and who will see it? Perhaps no one has put the point more forcefully than John Stewart Bell:
> What exactly qualifies some physical systems to play the role of ‘measurer’? Was the wavefunction of the world waiting to jump for thousands of millions of years until a single-celled living creature appeared? Or did it have to wait a little longer, for some better qualified system . . . with a PhD?
"reality" is just a picture painted on consciousness. A particle exists because consciousness, for some reason, perceive a particle. Physicists should stop looking for smaller particles, and look into space and time as something that emerge out of consciousness...
You are enthralled! "There is something there still!" you say to yourself.
But there is nothing there! The game engine just figured that the particle would collide with the triangle, and altered its velocity accordingly. But you are too caught up in your assumption that you do more experiments (I think this is what our physicists are doing in our world). But what ever you do, you can only observe the behavior implemented by the game engine, and not the game engine itself, or the computer that is running it....
But you can imagine it. You can imagine an explicitly implemented program and a computer with limited memory that is running it. And if you are allowed to consider it, they can do other experiments. Instead of looking into smaller and smaller things, you can look for other observations that might confirm this. Maybe you can go really far and see if they are not allowed to go beyond some threshold (Ref: The movie "The thirteenth floor").
And would that be pursuing something metaphysical?
> Would you father get hit with:
> * a 1 kg object moving at 1 m/s or
> * a 1 g object moving at 1 km/s?
And to illustrate the point a bit further, that same argument doesn't work in other speed/mass regimes. Would you rather get hit with:
- A 1000 kg object moving at 1 mm/s or
- A 1 kg object moving at 1 m/s
It's suddenly no longer intuitively obvious because both are totally fine. Like...roll a die, and my squishy human body will be equally happy with either result. Or, shifting the argument the other direction, would you rather get hit with:
- A 1000 kg object moving at 1 km/s or
- A 1 kg object moving at 1000 km/s
Pretty obviously the 1 kg object, despite its much higher kinetic energy. If I'm lucky it just takes off an arm or something, whereas the 1000 kg object is definitely splattering me.
And that sort of thing is why kinetic energy can be hard to reason about. Our intuition is calibrated around our squishiness, which only loosely relates to energy. There are other factors at play.
Why?
- Why more dangerous: A 1g steel ball is around 1/4 inch in diameter. A 1/4 inch hole in any major artery is an immediate life-threatening event, same if it hits the lungs, heart, or brain (gut-reaction: I think it'd be quite hard to penetrate the skull with this projectile, so to hit the brain you'd have to go through the eyes or pterion or something, but ignoring that sort of potential quibbling). It'll have pretty significant negative effects on any other system if it can actually pierce that system. If we pick a denser material like lead, the hole is smaller but still life-threatening. If we pick a less-dense material like wood, the hole is larger, and you're still pretty majorly screwed. Contrast that with the 1kg object, which is roughly equivalent to dropping two cans of beans from a height of 2 inches; it just doesn't matter.
- Why pierce: That's a big can of worms with a lot of answers. A halfway okay way to think about it for common hard materials and common shapes in the speed regime being discussed is that you have a higher energy to surface-area ratio.
Critically (in case you're using some sort of Socratic method to guide me to the increased energy being the rationale), the important number for piercing injuries is closer to being that ratio than the total energy, hence my example of 1mg at 1km/s in some recent comment. That has the same energy as our 1kg at 1m/s object, but it's still extremely dangerous because it physically punches out a 1mm hole through your body. If that strikes a location like your eye or a superficially exposed major artery (like the one on the inside of your elbow), you're still going to have a very bad day.
1 m/s is just 3.6 km/h, or slow walking speed. You'll be fine.
Your stupid brain thinks, oh I am just 10ish over the limit.
The exact numbers are more nuanced (both in simulation and in terms of available data on the topic), but to illustrate the effect, suppose typical reaction time allows you to slow to 0mph from 25mph just before striking a child with your car, assuming somebody with perfect training and a typical car applying proper emergency braking and decelerating at around 0.8 gs. I'll consider both the 0.5s reaction time estimate (median results in places where people are primed to have to react) or 1.5s (median for an alert driver expecting hazards occasionally). Let's assume we're going 10 over.
The fact that you were going faster means you cover more distance before you even hit the brakes. The 25mph traversed either 44.43ft in 1.92s (at 0.5s reaction time) or 81.1ft in 2.92s (at 1.5s reaction time). By the time the 35mph even applies the brakes, it only has either 18.77ft left or 4.1ft left (0.5s vs 1.5s reaction time).
Then how much can you slow down in that time? In the 0.5s reaction time case the impact speed is 27.85mph. In the 1.5s case it's 33.57mph.
In either case, you're actually striking the child above the posted speed limit in the same emergency-braking scenario, at a speed likely to cause significant injury.
Details obviously vary depending on how you set up the parameters, but 25mph is an approximate cutoff below which we empirically tend to have pretty safe traffic outcomes in residential areas and above which we tend to have pretty dangerous traffic outcomes. I don't know if people choose to speed in those situations because they don't know or because they don't care, but either way, it makes our roads much more dangerous than they need to be.
Instead of mph/kph the speed limit and speedometers could be on QuadraticSpeedUnits.
One can build new intuitions.
Axler's "Linear Algebra Done Right" https://linear.axler.net/LADR4e.pdf
Nielsen and Chuang's "Quantum Computation and Information" Chapter 2. (Available online, but legality depends on where you live)
If you're interested primarily in QM rather than linear algebra, I'd recommend not working through all of Axler, but instead using Nielsen and Chuang to learn about qubits and observables, and using Axler to fill in the gaps.
I loved it.
Quantum Processes, Systems, & Information by Benjamin Schumacher and Michael Westmoreland is another good treatment along those lines.