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Category Archives: 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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2 Comments | tags: 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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12 Comments | tags: linear algebra, matrix transform, 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, 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, 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

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.”*

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11 Comments | tags: Philip Ball, quantum mechanics, wave-function | posted in Books, Physics

Since I retired, I’ve been learning and exploring the mathematics and details of quantum mechanics. There is a point with quantum theory where language and intuition fail, and only the math expresses our understanding. The irony of quantum theory is that no one understands what the math *means* (but it works really well).

Recently I’ve felt comfortable enough with the math to start exploring a more challenging aspect of the mechanics: **quantum computing**. As with quantum anything, part of the challenge involves “impossible” ideas.

Like the square root of **NOT**.

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18 Comments | tags: Artur Ekert, Philip Ball, quantum computing | posted in Computers, Physics

With COVID-19 putting a damper on social activity, “the gang” doesn’t get together very often, but we still gather occasionally (and carefully). One of the times recently I got into how, even though we’re all sitting essentially motionless in a living room, we’re moving through *time* at the speed of light. I explained why that was, and they found it pretty cool.

Then I ran into someone online who just couldn’t wrap his head around it — just couldn’t accept it (despite explaining in detail and even providing some links). Physics is sometimes challenging to our daily perceptions of reality!

However in this case, it’s just a matter of some simple geometry.

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16 Comments | tags: frame of reference, light speed, Lorentz equation, spacetime, Special Relativity, velocity | posted in Physics

Last time I opened with basic **exponentiation** and raised it to the idea of complex exponents (which may, or may not, have been surprising to you). I also began exploring the ubiquitous *exp* function, which enables the complex math needed to deal with such exponents.

The *exp*(*x*) function, which is the same as *e*^{x}, appears widely throughout physics. The complex version, *exp*(*ix*), is especially common in wave-based physics (such as optics, sound, and quantum mechanics). It’s instrumental in the **Fourier transform**.

Which in turn is as instrumental to mathematicians and physicists as a hammer is to carpenters and pianos.

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8 Comments | tags: complex numbers, complex plane, exponential function, exponentiation, Fourier transform, Heisenberg Uncertainty | posted in Math, Physics