The word “always” always finds itself in phrases such as “I’ve always loved Star Trek!” I’ve always wondered about that — it’s rarely literally true. (I suppose it could be “literally” true, though. Language is odd, not even.) The implied sense, obviously, is “as long as I could have.”
The last years or so I’ve always been trying to instead say, “I’ve long loved Star Trek!” (although, bad example, I don’t anymore; 50 years was enough). Still, it remains true I loved Star Trek for a long (long) time.
On the other hand, it is literally true that I’ve always loved science.
I’ve mentioned before that my first two words (so I’m told) were “light” and “star” (while the star in question was on the top of a Christmas tree, I did go on to develop an early interest in astronomy — got a 3″ reflector telescope Christmas present in grade school).
As I wrote that I flashed on a distant and vague memory of an arranged nighttime visit to an observatory — my dad took me (someone he knew I think; a church member maybe). We looked through a big astronomer’s telescope at Saturn. But, maybe due to wearing glasses and being too young to know how to look, all I saw was a blur. And I was too shy to say anything or ask for help.
(Fortunately I was too young to be really heart-broken about not seeing Saturn. In my early 30s, I looked at Saturn through a friend’s very good reflector… and it was one of the most profound moments of my life. Photons from the Sun entangled with and bounced off Saturn and came all the way back to my eye and entangled with me. In a sense, I was physically connected with the planet. Then we looked at the moons of Jupiter, and there just really aren’t words for it.)
Anyway, as I say, I’ve always loved science. Literally.
I’ve nearly always loved physics — the foundations and basics of science. That doesn’t go quite as far back, but I cannot remember ever not being fascinated by how things work.
My father despaired at how I’d take things apart but never put them back together (which I never saw as the point). It wasn’t until much later that fixing things became interesting. Ironically, in my ultimate career I began as a Field Service Technician — the guy who showed up to fix your machine.
That career owes much to an early interest in electronics. Crystal radios in Cub Scouts. Shortwave radios in high school. In grade school I made shock boxes (high voltage, low current) to take to school for Show and Tell. (“Okay, everyone form a circle and hold hands…”)
I’ve been into this stuff since lasers were invented and quarks turned out to be real things.
It ramped up in high school when I subscribed to Scientific American and began reading science books. (I also subscribed to Playboy; I had very liberal and tolerant parents.)
Scientific American and I parted ways, but I’ve never stopped reading science books. Over time it escalated to science papers and blogs (and YouTube videos).
Since I retired in 2013, I’ve had time to get deeper into it, and recently I’ve become an “undergrad” for some quantum physics college courses. MIT has an Open Course Ware program that’s an amazing gift to autodidacts. (YouTube and Wikipedia, in general, are rich resources for science learning.)
The point of all this is that, after being a mathematical outsider to quantum physics, the reading and watching is paying off. I’m able to see deeper into the forest than I ever have before, and it’s a bit exhilarating.
It’s said quantum mechanics cannot be truly understood by language or analogy. The only real understanding comes from an appreciation of the mathematics. This… turns out to be true, and even the reality of this truth can’t be expressed, only experienced.
It’s a real-life version of Mary’s Room. Learning the math is seeing color for the first time. It almost makes certain puzzling things obvious (although not quite — quantum physics defies our intuitions regardless of perspective or knowledge).
Suffice to say there is a lot of, “Oh, that’s why…!”
There are topics I’ve actively studied for many years and about which I feel I can speak with some authority. Quantum physics isn’t going to be one of them for a long time. Or, frankly, maybe ever. (The higher math is where I need most growth. I can follow much of it, but not do it myself.)
Down the road I plan to write a post about experiments of Bell’s Inequality because, [A] it fascinates me, and [B] I’ve yet to see or read an account I really like. (While I really liked Philip Ball’s book, Beyond Weird, I disliked his analogy. I found it more confusing than useful, but it is a hard topic to explore. The challenge is finding a way to explain the math sensibly.)
There is also a [C] reason: In many ways, what we observe here is more astonishing than in the two-slit experiment. As astonishing, at the very least, but I think it’s a winner when it comes to shattering one’s intuition about reality.
What I think I might contribute now is exploring in detail some of the stumbling blocks I’ve struggled with. I can also enumerate some of the requirements — things one needs to know to learn quantum mechanics. And I can list some of the resources I’ve found helpful. (Certainly starting with MIT OCW. Can’t say enough about them.)
I’ll start here by touching on foundation topics necessary if one wants to get into a mathematical understanding of quantum mechanics. As it turns out, I’ve written about a number of them in the past. (Many of my different interests come together in QM. I never imagined exploring rotation matrices would be so useful here.)
Obviously one needs math skills; the pre-requisites are all math subjects. Luckily, the math for much of QM isn’t that bad, and it’s easier if one only wants to be able to follow along (my current tactic until/unless my math skills improve).
One absolutely will need:
In classical mechanics complex numbers are often used for convenience, but are either cancelled out in results or not actually necessary in the first place. Conversely, they are crucial and prominent in QM’s most famous equation.
If you don’t have a pretty good grasp of the complex numbers, you don’t have a prayer, so study up.
[see: complex numbers]
• The idea of a “space” with an arbitrary number of dimensions, in particular the idea of a vector space. If you’ve at least worked with an X-Y graph, you’ve gotten started. If you’re comfortable with 3D space and X-Y-Z coordinate systems, you’re on your way (but not there).
You’ll also want to be familiar with polar coordinate systems. Those become important for spin, but also feature in how quantum states are represented.
• Matrix math and linear transformations. Don’t let the titles intimidate, we’re talking about a really cool kind of geometry. As with all types of subjects there are some unfamiliar basics to learn at first, but then doors open.
Matrix math is just multiplying and adding. Its bad reputation comes from having to remember the order. And there’s a lot of writing, so high school students understandably really hate it. (Properly introduced, I think they’d find it fascinating.)
Linear transformations are vital because in QM they are linear operators, which represent observables and quantum basis states. In both cases they are (or can be) represented by matrices, which is where the matrix math comes in.
• Enough calculus to have a feel for derivatives and integrals. (This is a weak area for me. I can do basic derivatives and even more basic integrals, but have yet to get into more complex uses, let alone partial derivatives.)
To actually do the math, one needs to have a good grasp of calculus, but to follow along, a general understanding will get one through.
[see: Math Books (not that I’ve really posted on calc)]
The part I would guess that’s new for most is working with vector spaces and transformations. One of the MIT courses spent several lectures reviewing linear algebra as it applies to QM, and I found very helpful.
There is also a short tutorial at the Qiskit site. [From the site: Qiskit [quiss-kit] is an open source SDK for working with quantum computers at the level of pulses, circuits and application modules.]
Speaking of the MIT OCW courses, this is they:
I’ve been aware of the MIT courses for a while, so when I got serious about learning QM I searched for [MIT quantum physics] and a bunch of these popped up. They’re really good. Professor Adams is a high-energy delight to watch and clearly is having fun. Two thumbs up; highly recommended.
YouTube playlist (25 lectures, each about 80 minutes)
This is the same course taught a different year with a different teacher. I haven’t watched this series, but Professor Zwiebach is interesting enough that I might. (I’m definitely watching one of them again and maybe a third time as I get better and better at understanding the math.)
YouTube playlist (115 videos, but all quite short)
Professor Zwiebach spends the first nine or so lectures reviewing linear algebra (which is as far as I’ve gotten in the series). He has a different style than Professor Adams, but he’s also a very good and enjoyable lecturer.
YouTube playlist (26 lectures, each about 80 minutes)
There are many more. YouTube is a great resource!
And on (and because of) that rather lengthy note, I’ll end abruptly.
Stay quantum, my friends! Go forth and spread beauty and light.