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.
Initially I thought, for the first time in the the Brain Bubbles series, I have a bubble actually related to the brain. When I went through the list, though, I saw that #17, Pointers!, was about the brain-mind problem, although the ideas expressed there were very speculative.
As is usually the case when talking about the mind and consciousness, considerable speculation is involved — there remain so many unknowns. A big one involves the notion of free will.
I just read an article that seems to support an idea I have about that.
Trigonometry is infamously something most normal people fear and loath. Or at least don’t understand and don’t particularly want to deal with. (In fairness, it doesn’t pop up much in regular life.) As with matrix math, trig often remains opaque even for those who do have a basic grasp of other parts of math.
Excellent and thorough tutorials exist for those interested in digging into either topic, but (as with matrix math) I thought a high-altitude flyover might be helpful in pointing out important concepts.
The irony, as it turns out, is that trig is actually pretty easy!
There are many tutorials and teachers, online and off, that can teach you how to work with matrices. This post is a quick reference for the basics. Matrix operations are important in quantum mechanics, so I thought a Sideband might have some value.
I’ll mention the technique I use when doing matrix multiplication by hand. It’s a simple way of writing it out that I find helps me keep things straight. It also makes it obvious if two matrices are compatible for multiplying (not all are).
One thing to keep in mind: It’s all just adding and multiplying!
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.
If you know me, you know I’m not much a fan of official holidays, especially those that seem a bit on the artificial side (although National Waffle Day is something I would celebrate). That said, I’m not entirely untouched by a holiday that demands chocolates. Or roses, one of the very few flowers I do love. (Rose gardens are awesome.)
So I was delighted to see that, not only do Minnesotans have good sensibilities when it comes to voting politically, we also have good sensibilities when selecting our favorite Valentine’s Day candy.
See full map below. ION: Saw a cool new SF movie last night!
In what seems the distant past of late summer 2019, I posted about an interesting science fiction novel by Greg Egan, Quarantine (1992). The post didn’t get many views back then — only 13 that August, and only 27 total by the end of the year. And through 2020, it only racked up another 37 views. (That’s 64 total for those keeping score at home.)
Then, this January, the post got 257 views — 161 in the first three days. After being largely ignored for a year-and-a-half, something made the post go mildly viral. No one commented, so I have no idea how or why the post got so much traffic.
I have a thought it might have to do with the title.
I’m feeling outraged and depressed (because politics), plus it’s up to -2 outside (from -10 this morning), so I’m feeling very lazy about writing a post. (And I’m going to have to go out into the chill and shovel the light snowfall off my sidewalk and driveway. Brr!)
Therefore I’m offering up a lightly edited political piece I’ve had sitting in my folder of potential posts… since 2012. Which makes it both outdated and yet oddly still relevant. It’s a short piece, originally intended to be a Brain Bubble, but I’m just going to throw it out there as a regular post.
It’s a rumination on the differences between Left and Right in politics.
British author and philosopher Aldous Huxley blew my mind with what seemed like his incredible prescience in Brave New World (1932). In my post last December, thinking about our recent politics and social tone, I commented: “For a novel written 88 years ago, it’s surprisingly prescient and relevant.”
The novel impressed me so much I bought the series of essays Huxley published almost 30 years later, Brave New World Revisited (1959). So far, I’ve only read the first five (so many distractions these days), but the apparent prescience continues to astound and astonish me.
I qualify that with “apparent” because it’s actually as old as humanity.
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.