Classical and Quantum Mechanics

Hello Dear Readers,

If you've not read Euclid's Window, I highly recommend it. It's a look at the history of mathematics all the way from antiquity up through the 20^th-century development of String Theory.

My favorite part of the book is in Chapter 32 excerpted here:

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One night, after working on this chapter, the clash between Einstein and Heisenberg acted itself out in a dream. It all began with Nicolai, as Einstein, walking in and showing me some theories he had scrawled in crayon on his pre-K activity book...

Nicolai as Einstein: Dad, I've discovered general relativity! When matter is around, space is curved, but in empty space the gravitational field is zero, and space is flat. In fact, in any region that is small enough, space is approximately flat. (I'm about to say, "What a beautiful theory, can I hang it on the wall?" when Alexei enters.)

Alexei as Heisenberg: Sorrrrry. The gravitational field, like any field, is subject to the uncertainty principle.

Nicolai as Einstein: So?

Alexei as Heisenberg: So in empty space, while the field might be zero on the average, it is really fluctuating in space and time. And in really tiny regions the fluctuations are humongous.

Nicolai as Einstein (whining): But if the gravitational field is fluctuating, so is the curvature of space, because my equations show that the curvature is related to the value of the field...

Alexei as Heisenberg (taunting): Ha-ha! That means the space in tiny regions cannot be considered flat. ... In fact, when you look closer than the scale of the Planck length, tiny virtual black holes form.... It isn't very pretty...

Nicolai as Einstein: I said I want tiny regions of space to be flat!

Alexei as Heisenberg: But they aren't!

Nicolai as Einstein: They are!

Alexei as Heisenberg: Aren't.

Nicolai as Einstein: Are.

... In the dream this went on until I woke up palpitating. (I took it as a sign that I was not meant to sleep until I finished the chapter.)

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This was the first time I understood why Quantum Mechanics and General Relativity were incompatible. I wanted to summarize this for a more general audience, but after some back and forth with my editor I settled for summarizing Classical Mechanics and Quantum Mechanics instead, since an article on Relativity had already been written.

What Is Classical Mechanics?
http://www.livescience.com/47814-classical-mechanics.html

I initially began the story of classical mechanics just so I could refer to it by name when talking about QM. It ended up being a nice way to talk about the early days of the scientific method (15^th century) in order to contrast it to the way science (if it can be called that) was practice for roughly 5000 years since the invention of writing (34^th century BC.)

What is Quantum Mechanics?
http://www.livescience.com/33816-quantum-mechanics-explanation.html

The story of QM as it's usually told in undergrad textbooks strays from the true chronology in two separate places: when discussing the phenomenon of blackbody radiation, and the photoelectric effect.

The first thing that textbooks get wrong while Max Planck was certainly the first person use the notion of quantization when explaining the frequency distribution observed from the blackbody radiation, he regarded it as nothing more than a mathematical trick. Also, he wasn't at all concerned with the UV catastrophe of the Rayleigh–Jeans law. The problems Planck was actually concerned with had more to do with entropy and microstates. This doesn't fit very well into the narrative of quantum mechanics debunking classical mechanics, so I understand the allure of the traditional chronology of history. For more on this I recommend the following link:

http://physicsworld.com/cws/article/print/2000/dec/01/max-planck-the-reluctant-revolutionary

The second thing textbooks almost always get wrong is that Einstein wasn't quite justified in positing the existence of photons (which he called “light quanta”) for explaining the photoelectric effect. His 1905 paper, "Concerning an Heuristic Point of View Toward the Emission and Transformation of Light,” actually gave nine reasons for supposing the existence of photons, only one of which was the photoelectric effect. Certain semi-classical treatments of wave-nature of light are, in fact, perfectly suited to describing the phenomenon. His winning of the 1921 Nobel Prize was in controversial, and his award was specifically worded NOT to rely on any notion of photons: “for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect.” (The Nobel Committee was fairly skeptical of Einstein’s Relativity as well.) Compton scattering, where light scattered by an electron beam changed in frequency, was actually the first clear evidence for the correctness of Einstein’s hypothesis of the existence of photons. For more on this I recommend the following link:

http://sci-ed.org/documents/Klassen-Munich.pdf

I really enjoy the challenge of representing history accurately to as diverse and as wide audience as possible. I hope my readers enjoy it.

Proofs: Extra Stuff

New article up at LiveScience:
http://www.livescience.com/46254-proof-theorem-axiom.html

And as bonus, here's the story of what prompted me to write it.
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Thinking back to my Middle School days (only briefly... I can laugh now about how mean kids are when they're learning how to become people, but only because those days are long gone) I have this memory of chatting with my brother on the way back from the bus stop.

"Oh, and we learned about the Pythagorean Theorem in math today," he mentioned. I'd had a couple years more math than him at this point (and to me it was a bit of a hobby). Out of curiosity I asked:
"Did you prove it in class?"
"Yeah."
"Really? Which method did you use?"
"Uh... I don't know how to explain this without drawing a picture." The streetside upon which we were walking was covered in bark-o-mulch, so I implored him to humor me and use the ground as a canvas. He pretty much drew me this:

At the time, I lacked the words (and probably civility with either of my siblings) to say "No, that's not a proof." Here's the conversation I would have liked to have:
"How is this not a proof? I just showed the Pythagorean Theorem works!"
"You've shown the Pythagorean Theorem works in ONE case. The reason a statement like the Pythagorean Theorem is so powerful is because we know it works for EVERY right triangle."
"How am I supposed to draw every right triangle?"
"You can't. That would literally take forever."
"...How did Pythagoras prove it then?" I, in turn (and again in the mulch), would have drawn something like this:


"The square that forms in the middle space happens to be the square of the triangle’s hypotenuse (c²) . If we rearrange two of the triangles, we see that two squares are created, each one having the area of the square of each of the triangle’s legs (a² + b²). Since the two areas have to equal each other, we see that a² + b² = c²."
"But how's that better than the diagram I just drew?"
"Because this one works for EVERY right triangle."
"And mine doesn't?"
"The way you did it, we have to measure the sides to make sure it works, and repeat that measurement for every possible right triangle. With the way I just did it, we don't have to measure the sides of the triangle. Draw a right triangle in your head. It can be as big or small as you like, and the legs can be as close or as disparate in length as you like. I don't even have to see it. I know it will work with the procedure I just drew. The space in the middle of the square is always c², and the rearrangement always produces a² + b², and we know that the two areas always have to be equal."
"...And every proof is like this?"
"This is why mathematics is special apart from other sciences. What you drew would be analogous to an 'experiment.' The best we can do for scientific claims is experiment in order to verify or debunk them. If we try our hardest to debunk a claim using controlled, rigorous experiments and repeatedly and consistently FAIL, then we have no other choice than to accept that claim as true. This is what we mean by a good scientific claim being 'falsifiable.' It might be impossible to prove the earth is held by a giant invisible octopus, but there's no way to test such a claim. A claim like "lightning is a form of electricity" not only explains certain things we know about lightning, but we've been unable to produce experimental results that run contrary to that claim."
"But experiments aren't enough for mathematical claims?"
"No, because we can do better than experiment; we have proofs. If we suspect something to be true in mathematics before it's proven, we call it a conjecture."
"How's that different from a hypothesis?"
"A scientific claim makes the transition from hypothesis to theory as it stands against some duration of controlled rigorous experimentation. A mathematical claim makes the transition from conjecture to theorem once a proof is thought up. It doesn't matter if a conjecture stands the test of time. Until somebody thinks up a proof, it will remain a conjecture forever. One example is the "Twin Prime" conjecture. Mathematicians have believed since antiquity that there's an infinite number of pairs of prime numbers that have a difference of two, but nobody has been able to think up a proof."

That all aside, as first-year grad students, most members of my PhD chemical engineering class thought that the way to prove a real symmetric matrix is diagonalizable was to punch in numbers and see if it works... meaning none of them understood what a mathematical proof is either. I've not figured out why this isn't public knowledge, as it's the fundamental difference of why math is special apart from the other sciences.

So there. Two reasons why I thought the article was worth writing.

What is a Transistor?: Extra stuff

If you've ever wondered what a transistor is, I've done my best to explain over at LiveScience: http://www.livescience.com/46021-what-is-a-transistor.html

I originally planned to talk more about how transistors enable memory. Here's all that made it:

Through looping signals backwards, certain kinds of memory are made possible by signal-triggered switches as well. While this method of information storage has taken a back seat to magnetic and optical media, it is still important to some modern computer operations such as cache.

The notion of "looping signals backwards" is the basis of what electrical engineers call "flip flops" or "latches". Wikipedia has a neat article on these, but it won't mean a lot unless you're familiar with how transistors or logic gates work: http://en.wikipedia.org/wiki/Flip-flop_(electronics)

Here's how I first understood SR latches:

Here signal S (“Set”) will turn the light on, and it will remain on after S is turned off (or even turned on/off again). Similarly, signal R (“Reset”) will turn the light off, and it will remain off after R is turned off (or even turned on/off again). This setup stores only one bit, but more relays can be used to store more information.

Sometimes this is called an "SR NOR latch" because it behaves the same way as two NOR gates wired to loop back on each other. Wikipedia has a nice animation of how this works:

(credit Wiki User Napalm Llama, 2008)

For a long while, I thought this is how flash memory worked... but this was due to misunderstanding synonyms used in various areas of electrical engineering ("NOR" and "gate" in particular). Flash memory is actually enabled through special kinds of field-effect transistors, but the one of the terminals (the gate) is constructed a little differently from a traditional FET; said to be "floating". The details of how this enables the storage, accessing and erasure of memory still confuses me. http://en.wikipedia.org/wiki/Flash_memory

That's all for now. I hope this was interesting.

-Robert

Keeping Time: Why 60 Minutes?: Maps of Civilization

Over at LiveScience I've been writing a series on all the ways me measure time in modern times, specifically on all the names, numbers, and conventions that we associate with each:

• Years (AD/BC, and why there's no year zero)
• Leap Years (How they work and the history of the Gregorian Calendar)
• Months (Why there's 12, their names and lengths)
• Weeks (Why 7 days, names of the days)
• Hours (Why there's 24, why clocks only show 12, and time zones)
• Minutes and Seconds (Why 60?)

By far, my most favorite is the last one, because it's basically a history of civilization up until the scientific revolution: http://www.livescience.com/44964-why-60-minutes-in-an-hour.html

The best thing about this story is the maps. It's so hard to talk about the progress of knowledge before the scientific revolution because it spans the lifespan of so many civilizations. To me the key was looking at the maps and actually being able to see knowledge flow and grow over history. These maps are actually what inspired me to tell the history of sexagesimal as a single, unbroken story spanning the entire history of civilization.

Sexagesimal was invented in the 34th century BC with the first writing. It was the way all numbers were recorded for two thousand years.

Continuing to write in sexagesimal, the Babylonians also invented the degree. Most importantly, the Babylonians applied these principles to observing the sky, and took measurements that constituted the first study of astronomy. Their work would serve as the basis for another thousands years, in spite of the boom and bust of many great empires.

Alexander the Great's conquest reached India in 326 BC, spreading the knowledge of Babylonian astronomy to both Greece and India. This work created such a strong association between sexagesimal and astronomy that even though the Greeks had their own system of numerals in decimal, Greek (and later Roman scholars) kept using sexagesimal for star charts, trigonometry, and navigation.

Much of this knowledge was lost to Europe for many centuries, beginning with the fall of the West half of Rome in the 5th century AD. The Islamic-Arabian empires inherited many Roman (and later Indian) ideas starting with the Rashidun Caliphate in the seventh century. Muslims scholars, after expanding on this knowledge greatly, reintroduced it to Europe in the eighth century through the Iberian Peninsula, which was then part of the Umayyad Caliphate.

The knowledge that had been saved and enhanced by Muslim scholars found its way to Christian scholars in the 11-13th centuries. Europe and the Islamic lands had multiple points of contact during the Middle Ages. Particular points of transmission of Islamic knowledge to Europe were:

• The 1085 conquest of Toledo by Spanish Christians
• The 1091 re-conquest of Sicily by Normans (following Islamic conquest in 965)
• The Crusader battles in the Levant (1096–1303)

During the 11th and 12th centuries, many Christian scholars traveled to Muslim lands to learn sciences. Notable examples include:

• Leonardo Fibonacci (1170 – 1250)
• Adelard of Bath (1080 – 1152)
• Constantine the African (1017 – 1087)

Some other graphics I put together for each civilization:

Sumer & Akkadia:

Babylonia:

Greece & Rome: