Category Archives: Science
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.
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40 Comments | tags: QM101, quantum mechanics, quantum spin | posted in Math, Physics
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.)
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14 Comments | tags: eigenstate, eigenvalue, eigenvector, matrix transform, QM101, quantum mechanics | posted in Math, Physics
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.
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6 Comments | tags: QM101, quantum mechanics, quantum operator | posted in Math, Physics
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!
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31 Comments | tags: cosine, sine, sine wave, trigonometry | posted in Math, Sideband
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!
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7 Comments | tags: matrix math, matrix multiplication | posted in Math, Sideband
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.
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13 Comments | tags: linear algebra, matrix transform, QM101, quantum mechanics, vector space, vectors | posted in Math, Physics
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.
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9 Comments | tags: coordinate system, inner product, QM101, quantum mechanics, vector space, vectors | posted in Math, Physics
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.
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12 Comments | tags: MIT OCW, QM101, quantum computing, quantum mechanics, Scientific American | posted in Physics
A crushed flower.
This post has nothing to do with Amy Winehouse, sadly on the list of great talents who, poorly served by those in their lives, lost their way and died tragically and long before their time. (It’s bad enough when the ravages of life — disease and accident — steal away those with gifts. Losing people to human foibles is a more painful loss.)
The topic here is the Block Universe Hypothesis, which I’m revisiting, so the title kinda grabbed me (and I am a Winehouse fan). I’ve written about the BUH before, but a second debate with the same opponent turned up a few points worth exploring.
So it’s back to basic block (everyone looks good in block?)…
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31 Comments | tags: block universe, simultaneity, Special Relativity | posted in Physics
Among those who study the human mind and consciousness, there is what is termed “The Hard Problem.” It is in contrast to, and qualitatively different from, problems that are merely hard. (Simply put, The Hard Problem is the question of how subjective experience arises from the physical mechanism of the brain.)
This post isn’t about that at all. It’s not even about the human mind (or about politics). This post is about good old fundamental physics. That is to say, basic reality. Some time ago, a friend asked me what was missing from our picture of physics. This is, in part, my answer.
There is quite a bit, as it turns out, and it’s something I like to remind myself of from time to time, so I made a list.
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26 Comments | tags: big bang, block universe, cosmology, dark energy, dark matter, Many Worlds Interpretation, Mathematical Universe Hypothesis, MUH, MWI, quantum physics, standard model, universe, virtual reality | posted in Physics
Logos con carne 