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:
• The complex numbers and the complex plane. The quantum world depends on the complex numbers; they are what makes QM different from classical mechanics. They are fundamental to the math.
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.
[see: Dimensional Coordinates, Vectors and Scalars (oh, my!) & configuration-space]
• 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.
[see: matrix-math]
• 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:
Quantum Physics I (course #8.04; spring 2013; Prof. Allan Adams)
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)
Quantum Physics I (course #8.04; spring 2016; Prof. Barton Zwiebach)
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)
Quantum Physics II (course #8.05; spring 2013; Prof. Barton Zwiebach)
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)
§
Here’s another playlist of videos by an interesting lecturer. His enunciation is fascinating. His channel, AK Lectures, has a lot of other lecture playlists.
I’ve only seen a couple videos in this series by Dr. Underwood, but they were good enough I saved the playlist. His channel also has more playlists.
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.
∇
Logos con carne 
February 1st, 2021 at 7:56 am
I have no set schedule or agenda for these “QM 101” posts, but topics I’m already working on include vector spaces and eigenvectors (and eigenvalues), the wavefunction, and quantum states. The intent is to help those, like me, who are interested in getting into QM through the math door.
February 1st, 2021 at 5:28 pm
That’s a lot of math prereqs. While I have the most basic (and in many cases very rusty) understanding of many of those areas, I can’t say I’m proficient in any of them. But then, my dips into QM math have really been just the bare minimum necessary to understand some of the assertions being made by various physicists. But I doubt I’ll ever acquire enough to actually work with it.
That said, I’m still very much looking forward to those posts!
One thing I haven’t been able to wrap my brain around, is what it means to use a bra in bra-ket notated expression. I understand it’s a row vector as opposed to a column vector in a ket. What I don’t get is why you’d use them, as opposed to just different kets multiplied together.
I saw a couple of the Allan Adams lectures and was impressed. Everything made sense until the math. He made it clear he was tapping into concepts from a prereq course the students had taken, so I’m sure they understood what he was talking about, but for me it was lots of gobbledygook with an occasional point I understood (such as calling the non-linear examples, like his signature, “stupid”).
Anyway, those look like some excellent playlists!
February 1st, 2021 at 8:06 pm
“I can’t say I’m proficient in any of them.”
Obviously it depends on your interest, but as mentioned in the post, the complex numbers and complex plane are fundamental (including the e-to-the-i-x stuff I was talking about recently). Everything else builds on that, so it’s the big pre-req to everything else.
I doubt I’ll ever work with the math, either. Not seriously, anyway. I’d like to be able to do the calculations for some simple Schrödinger situations and make some images or animations. If mine turn out like others I’ve seen, it would demonstrate I’m getting it right.
“What I don’t get is why you’d use them, as opposed to just different kets multiplied together.”
The short answer is that ⟨b|a⟩ gives the inner product of vectors a and b. One use for this is in determining orthogonality. Given:
Then:
And:
Which means |0⟩ and |1⟩ are orthogonal (as expected) and, of course, |0⟩ is not orthogonal to itself. (In fact, the 1 means it’s equal to itself, which it has to be.)
That Professor Adams was referencing stuff the students were expected to know has been one of the challenges. I’ve been detouring while I tried to pick up some of that background. Over time and with repetition I’m picking up stuff. I plan to watch the lecture series again; I know a lot of what went over my head the first time will fly by a lot closer this time.
“(such as calling the non-linear examples, like his signature, ‘stupid’)”
Heh, yeah, I remember that. The wave-function is something we define, and there are a number of constraints on what’s allowed. They boil down to the definition not being “stupid” though. 😀
February 1st, 2021 at 9:02 pm
Thanks for the explanation on orthogonality. That matches what I’ve read. What I’ve struggled to find an explanation of is what it means in terms of the physics. Most wavefunctions I see laid out just have kets, something like:
c1|ket1> + c2|ket2>
But occasionally I see something like:
and I don’t know what it signifies, aside from involving the inner product of orthogonal vectors.
On Adams and the prereqs, yeah, the lecture I was watching was from 8.04, intro to QM, but it had a prereq of 8.03, vibrations and waves, which I just realized is online but in the classical physics section. I might poke around in that one to see if it makes 8.04 a little less opaque. Although I suspect I’ll then have to look at 8.02. 😛
February 1st, 2021 at 9:23 pm
(Your second example went to bit heaven somehow, but I assume it was some version of ⟨Ψ|Φ⟩. FWIW, you can use the HTML codes ⟨ and ⟩ for the angle brackets.)
The form:
Ψ = α|ψ_1⟩ + β|ψ_2⟩
Represents a linear superposition of ψ_1 and ψ_2 with coefficients α and β, respectively. Note that we can add and subtract kets, but not multiply them (when they’re column vectors, anyway; one cannot multiply column vectors).
The inner-product form:
n = ⟨a|b⟩
Is a completely different operation. Orthogonality is just one aspect of it. It’s also how we determine expectation values — the expected value of a measurement.
Specifically, ⟨a| is the conjugate transpose of |a⟩ — a matrix operation that flips the matrix on its identity diagonal and takes the complex conjugate of each member. That’s how a column vector |a⟩ becomes the row vector ⟨a*| (a* is the complex conjugate of a).
There is also an outer-product:
m = |b⟩⟨a|
Which creates a square matrix from the vectors |a⟩ and |b⟩ — the operation can be used to define operators in terms of quantum states.
February 1st, 2021 at 9:27 pm
Does it help if I add that taking the inner product guarantees a real valued answer (not a complex value), and an inner product is always a scalar, not another vector. Adding kets in superposition always gives a new ket, a new wave-function. The inner product is a way of taking the momentary value of a wave-function.
February 2nd, 2021 at 10:25 am
Thanks. Yeah, WP eating anything that resemble HTML it doesn’t know about is annoying, and why I rarely bother to try putting things in Dirac notation. Do you use something to code that stuff, or just hand code the html?
I do often see what looks like kets multiplied together, although I think it’s actually an implied tensor product. For example: https://www.askamathematician.com/2018/10/q-what-is-the-monogamy-of-entanglement/ (You’ll have to scroll a bit to see the notation)
Actually the distinction between inner and outer product does help. So when they’re coded in that manner, a scalar value is being produced. That’ll at least give me a clue on what’s different the next time I see it. Thanks!
February 2nd, 2021 at 1:52 pm
On the one hand, it’s nice that WP comments support HTML, but on the other hand, it does lay some mines one has to avoid stepping on. One that often bites me is the ampersand. I often forget to type & instead. Fortunately, I’m not prone to using & in the first place.
Properly speaking (and I’m sure you already know this), the less-than and greater-than symbols are the wrong angle brackets for bra-ket notation. The preferred notation is the HTML ⟨ and ⟩ codes, ⟨ and ⟩, with the standard vertical pipe symbol.
There’s a cheat if you spend a little time using LaTeX, since WP supports it. The basic form is ＄latex code＄ where code is the LaTeX code.
This can be embedded in text, as in
or 
, or done in standalone paragraphs:
(There are also codes for setting image size and color. See this WP support page for details.)
Hopefully I got the above right. I’ll find out when I submit the reply! I often write replies off-line in my Windows based version of gvim, which is the GUI version of vim, which is the improved version of vi, which is the Unix editor I learned to love. It gives me HTML syntax highlighting, which helps with the HTML, and I have a variety of macros set up to make life easier. I copy-n-paste into the WP comment box.
I’ve used HTML since the early 1990s, so I’m pretty comfortable with it. I have some tools I made to help. This Unicode browser, this list of named entities, even this color-picker, handy tables,… (Everything written in good old DHTML!) 😀
“I do often see what looks like kets multiplied together, although I think it’s actually an implied tensor product.”
Yes, exactly. Matrix multiplication rules forbid multiplying two column vectors. Combining quantum states (as opposed to superposing them) requires a tensor product. There are various ways to write it:
Note that tensor products don’t commute:
“So when they’re coded in that manner, a scalar value is being produced.”
Just to be clear, the result of an inner product is a scalar. The result of an outer product is a square matrix.
(Okay, now to see if all this fancy stuff posts correctly…)
February 2nd, 2021 at 5:18 pm
Well done! And thanks.
February 2nd, 2021 at 5:29 pm
You’re quite welcome!
February 8th, 2021 at 7:48 am
[…] quantum states, eigenvectors and eigenvalues, the Bloch sphere, and spin. As I mentioned in the Introduction, the ultimate goal is exploring the Bell’s Inequality […]
February 15th, 2021 at 7:49 am
[…] mentioned in the Introduction that, despite the unfamiliar name, linear transformations — using linear algebra — are just a […]