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Category Archives: Physics

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.

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5 Comments | tags: bra-ket notation, inner product, matrix multiplication, outer product, QM101, quantum mechanics | posted in Math, Physics

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.

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3 Comments | tags: Bloch sphere, QM101, quantum computing, quantum mechanics, quantum spin | posted in Math, Physics

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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21 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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13 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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4 Comments | tags: QM101, quantum mechanics, quantum operator | posted in Math, Physics

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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28 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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23 Comments | tags: big bang, block universe, cosmology, dark energy, dark matter, Many Worlds Interpretation, Mathematical Universe Hypothesis, MWI, quantum physics, standard model, universe, virtual reality | posted in Physics