When I was in high school, bras were of great interest to me — mostly in regards to trying to remove them from my girlfriends. That was my errant youth and it slightly tickles my sense of the absurd that they’ve once again become a topic of interest, although in this case it’s a whole other kind of bra.
These days it’s all about Paul Dirac’s useful Bra-Ket notation, which is used throughout quantum mechanics. I’ve used it a bit in this series, and I thought it was high time to dig into the details.
Understanding them is one of the many important steps to climb.
One small hill I had to climb involved the object I’ve been using as the header image in these posts. It’s called the Bloch sphere, and it depicts a two-level quantum system. It’s heavily used in quantum computing because qubits typically are two-level systems.
So is quantum spin, which I wrote about last time. The sphere idea dates back to 1892 when Henri Poincaré defined the Poincaré sphere to describe light polarization (which is the quantum spin of photons).
All in all, it’s a handy device for visualizing these quantum states.
Popular treatments of quantum mechanics often treat quantum spin lightly. It reminds me of the weak force, which science writers often mention only in passing as ‘related to radioactive decay’ (true enough). There’s an implication it’s too complicated to explain.
With quantum spin, the handwave is that it is ‘similar to classical angular momentum’ (similar to actual physical spinning objects), but different in mysterious quantum ways too complicated to explain.
Ironically, it’s one of the simpler quantum systems, mathematically.
Unless one has a strong mathematical background, one new and perhaps puzzling concept in quantum mechanics is all the talk of eigenvalues and eigenvectors.
Making it even more confusing is that physicists tend to call eigenvectors eigenstates or eigenfunctions, and sometimes even refer to an eigenbasis.
So the obvious first question is, “What (or who) is an eigen?” (It turns out to be a what. In this case there was no famous physicist named Eigen.)
In quantum mechanics, one hears much talk about operators. The Wikipedia page for operators (a good page to know for those interested in QM) first has a section about operators in classical mechanics. The larger quantum section begins by saying: “The mathematical formulation of quantum mechanics (QM) is built upon the concept of an operator.”
Operators represent the observables of a quantum system. All measurable properties are represented mathematically by an operator.
But they’re a bit difficult to explain with plain words.
Last time I set the stage, the mathematical location for quantum mechanics, a complex vector space (Hilbert space) where the vectors represent quantum states. (A wave-function defines where the vector is in the space, but that’s a future topic.)
The next mile marker in the journey is the idea of a transformation of that space using operators. The topic is big enough to take two posts to cover in reasonable detail.
This first post introduces the idea of (linear) transformations.
Whether it’s to meet for dinner, attend a lecture, or play baseball, one of the first questions is “where?” Everything that takes place, takes place some place (and some time, but that’s another question).
Where quantum mechanics takes place is a challenging ontological issue, but the way we compute it is another matter. The math takes place in a complex vector space known as Hilbert space (“complex” here refers to the complex numbers, although the traditional sense does also apply a little bit).
Mathematically, a quantum state is a vector in Hilbert space.
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 think we all agree 2020 has been, as the curse puts it, an “interesting” year. Going into it, I had intentions about making changes. Most fell by the wayside due to COVID-19; I still haven’t taken the bus to watch the St. Paul Saints play. Or the bus-light rail combo to Target Field.
As a life long hard-core introvert, “social isolation” mostly meant I shopped for groceries less often but stocked up more when I did. The pain was fewer occasions of meeting a friend for tasty food, drink, and chat. I’m really looking forward to dining out again.
All-in-all, the last four years, this year… It’s been exhausting.
I just finished reading Beyond Weird: Why Everything You Thought You Knew About Quantum Physics Is Different (2018) by science writer Philip Ball. I like Ball a lot. He seems well grounded in physical reality, and I find his writing style generally transparent, clear, and precise.
As is often the case with physics books like these, the last chapter or three can get a bit speculative, even a bit vague, as the author looks forward to imagined future discoveries or, groundwork completed, now presents their own view. Which is fine with me so long as it’s well bracketed as speculation. I give Ball high marks all around.
The theme of the book is what Ball means by “beyond weird.”