Quantum Tunneling: A Different KInd of Calculation

I did some interesting physics yesterday so I thought I should write it up. The question was about quantum tunneling. It came from my friend's 3rd year Physical Electronics course in Electrical Engineering. You had a MosFET with a silicon dioxide insulating layer between the gate and the channel. They gave you the the barrier potential in the insulator, and told you to assume an electron with a certain energy (less than the barrier potential) in the gate. What was the "probability" or rate of tunneling?

This is a pretty standard problem which you can look up on Wikipedia. You calculate the form of the free-electron solutions in all three regions and match up the boundary condition. It's a little messy but it works.

I thought I could do better. The problem is messy because the conditions are different on the left and right hand sides. So I came up with the idea of working with symmetries. Instead of taking a freely propagating wave from the left and following it through the barrier, I put the whole thing inside a bigger potential well and considered the steady state solutions. Of course there are two minimum-energy solutions: the symmetric and the anti-symmetric, separated by a very tiny energy difference.

I don't like to carry too many letter symbols so I picked numbers that would come out nicely. From the terms of the problem, it's easy to calculate the free-propagation constants in both regions. So I allowed myself to tweak the problem parameters so the propagation constants come out to nice integers. Assuming pi = 22/7, here is the problem as I set it up:

You see I gave the well a width of 20 (you can call it 20 Angstroms if you like or 20 millimeters, it won't matter in the end...it's all about the geometry.) Can you see why this fits the sine wave perfectly? It's because the wave penetrates into the barrier. I've assumed that the penetration length is 2, so 20+2=22 and everything fits. The anti-symmetric solution is the same basic idea:

You can see I set it up so that there is a wavelength of 44, which fits perfectly into the box. But actually that's a bit of a cheat. If I were matching up a sine wave to an exponential at the boundary, then I would indeed get a wavelength of exactly 44, because of the well-known property of an exponential whereby the projection of the slope intersects the x-axis at exactly the decay length (I'm also assuming the sine wave is in the small-angle regime where sin(x)=x):

I don't really care about the constant in front of the sine wave, but I just threw it in here to show that it's easy. I used the property of sine waves that the projection of the tangent from the zero-crossing reaches the altitude of the sine wave after one radian. The point is that if I'm matching my sine wave (period = 44) to an exponential (decay length = 2) then this is how they line up. Everything fits.

Except I'm not exactly matching the sine wave to an exponential. I'm matching it to cosh (in the symmetric case) and sinh (in the antisymmetric case. Cosh is just a little higher and a little less steep...so it matches the sine wave at a slightly different position. Actually...the only way to match it up is to make the sine wave just a little bit longer. Instead of a half-length of 22 angstroms, the sine wave stretches to around 22.04....close to 0.2%, if you like. Similarly, in the anti-symmetric solution, the sine wave has to be shortened by the same amount. So the anti-symmetric solution has a slightly higher frequency than the symmetric solution.

We'll see what the implications of this are when we come back.

Oh...how did I calculate the 0.04 angstroms? That's the beauty of this method...it's pure geometry, just looking at the ration of the function with its derivative on both sides of the boundary, and using the small-angle approximation tan(x) = x. It falls right out.

Wave Function Collapse Explained by Quantum Siphoning

It's almost five years since I posted my original article on Quantum Siphoning and I think it's time for a second look. The idea is to explain the collapse of the wave function by means of normal time evolution. Specifically, how does the very weak "wave function" of the light from a distant star induce the reduction of silver bromide ("collapse of the wavefunction") on a photographic plate?

The traditional explanation is that this phenomenon proves the existence of photons, because the energy of the wave has to be concentrated in a point in order to provide:

a) the positive energy needed to account for the energy difference between silver bromide and metallic silver; and,

b) the additional energy needed to overcome the "bump" of promoting an electron into the conduction band.

The traditional picture looks like this:

 And they say you can't explain this with classical electromagnetism because the classical wave is far to weak to concentrate enough energy in that one little silver atom. Hence the photon.

I don't buy it. The first problem with this argument is that the thermodynamics is flawed. The photographic plate is not a device which captures energy and converts it to chemical form. The energy is already present in the plate in chemical form. Yes, it's true that the standard enthalpy of metallic silver is greater than that of silver bromide. But to calculate the spontaneity of the reaction you need to take into account the concentrations. The silver concentrations in an exposed photographic plate are parts per trillion. At those concentrations, the free energy of the reactions actually tilts the other way! I've done the calculation here. You don't need the energy of the "photon" to drive the reaction. The energy is already available in the chemistry of the plate.

But what about the "activation energy"...the bump of energy needed to get the electron into the conduction band, the intermediate stage of the process? That's where Quantum Siphoning comes in. The energy released when the target silver atom gets reduced is pumped back into the crystal to break up the silver bromide bonds. But how can it do that?

I couldn't figure this out for the longest time because I was trapped in the paradigm of a single electron getting transferred from one site in the crystal to another. But that's not what electrons are. They aren't particles with their own distinct identities. They are a collective wave function with multiple excitations. Anyone who knows about quantum field theory knows this is true.You simply can't describe an atom with two electrons by saying "this electron is here and that one is there". They are both excitations of a single wave function.

And the funny thing is, the same is true on some level for the x-trillion silver atoms which are part of a single silver bromide crystal. You can't say that a photon comes along and knocks an electron out of a single silver atom. What you can say is that a wave passes through the crystal and disturbs the wave function of the whole crystal so it is driven, ever so slightly, into a superposition of states where there is some amplitude that "an electron" is in the conduction band.

This is where people have trouble understanding where I'm going next. And the reason they have trouble is the way quantum mechanics is taught, from a strictly Copenhagen perspective of particles and quantum leaps. Nobody tells them about the equally-valid Schroedinger picture of wave functions and time evolution, which gives exactly the same results for all kinds of ordinary things, including the black body spectrum, the photo-electric effect, and even the Compton effect. I explain the connection between the two pictures in a series of blogposts starting here.

What people don't understand is that everything an atom does in its interactions with ordinary thermal light can be understood by looking at the superpositions of states calculated by the Schroedinger equation, and applying the charge density interpretation to the resulting wave functions, instead of Copenhagen's "probability density". People don't know this!

But it's much worse than that. They certainly don't know the full implications of the wave function picture, as I've listed two paragraphs back. That is not so surprising. What is horrifying to me is that they don't even know the basic and obvious fact that is you take the superposition of the s and p states of a hydrogen atom, you get an oscillating charge density...in other words, and antenna.

They don't know this...and when I tell them, they don't believe it. Even though its an obvious consequence of the well-known solutions of the Schroedinger equation.

And if they don't believe that the Schroedinger equations gives you a hydrogen atom with an oscillating charge distribution, then how are they going to believe that the very same tiny oscillating charge behaves exactly like a classical antenna? That's the calculation I did in those articles I pointed out above, comparing the Copenhagen picture to the Schroedinger interpretation.

And yet there is nothing that should be terribly controversial about anything I have said so far in this article. It's certainly unfashionable to talk about charge densities instead of probabilities, but there is nothing objectively wrong with it. What is appalling to me is that it is so very unfashionable that educated physicists scoff at the very notion that applying Maxwell's equations to the charge density picture gives you correct quantum-mechanical results. But it most certainly does.

So how does all this apply to the photographic plate? It's really very simple. The trillion-or-so silver atoms in the silver bromide crystal are little receiving antennas. The presence of a very week electromagnetic wave drives them ever so slightly into the excited state. What state is that? It's a state where the conduction band is ever-so-slightly excited.

Now there is one particular silver atom which is the target site, the site which we hope to convert to metallic silver. The presence of charge density in the conduction band will naturally couple to this target atom. And this coupling turns that special site into...an antenna. Just like the trillion little silver bromide antennas...but with a difference. Those silver bromide sites were receiving antennas. This target silver atom is...a transmitting antenna. Just as an atom being driven from a lower energy state to a higher energy state functions as a receiving antenna, so does an atom capturing an electron from a higher energy state into a lower energy state function as a transmitting antenna. I explain how these things work in this blogpost about the Crystal Radio.

The very weak electromagnetic wave is gone. It disturbed the silver bromide crystal to a very tiny extent, leaving a small amount of energy distributed among the ground state (silver bromide) and the two other states...the conduction band and the reduced silver site. And the combination of those two states creates a tiny classical antenna...a transmitting antenna buried in the middle of that micro-crystal. And now Quantum Siphoning takes over.

I always knew that the target silver atom would re-transmit energy at the optical frequency. I just never knew how the silver bromide "molecule" could re-capture that energy, which was going out in all directions. Until I realized it's not a single silver-bromide site which is involved...it's the whole silver-bromide crystal. Each silver-bromide site within that crystal is a receiving antenna, and those receiving antennas surround the target silver atom, which is transmitting. It's a perfect classical siphon. As the electron amplitude flows into the target silver atom, the conduction band is being depleted. But the energy released at the target site, in the form of re-radiated e-m waves, is captured by the surrounding silver-bromide sites...thereby replenishing the conduction band.

There is no photon. There is no collapse of the wave function. There is nothing but the ordinary, natural time-evolution of the Schroedinger function, working together with Maxwell's equations.

It's a perfect classical siphon.

What's wrong with the education system

When I was a grad student twenty-some years ago, my profs knew me as a guy who liked to come up with his own way of analyzing things. My thesis advisor was in Electrical Power, and everything there is done by "modelling". You have a motor or a transformer, and there is a "model" which represents its internal parameters like "core losses" or "magnetizing inductance". You do some external measurements from which you calculate those parameters, and then you analyze the machine as a simple circuit using the parameters you just calculated.

I never did this. I always worked from physical logic and analyzed things from the ground up. Once my advisor asked me, "Marty, why don't you like models?" (He was Rob Menzies, actually a very capable engineer and a pretty good prof.) I told him I didn't like them because they encouraged you to work by numbers without actually understanding what you were doing. I don't know if he got my point, but the other day I had a flashback to that moment.

I was working with some Engineering students the other day and one of them asked me a question from his Power Systems course. It was about transformers. You do some measurements on the transformer and calculate its equivalent circuit parameters. I asked him to show me the question, and he did. It started off something like this:

"You have a single-phase transformer rated 20 kVA, 2200V primary and 220 secondary. You do an open circuit test and measure 220V, 2.5 A and 100 Watts. Then you do the short circuit test and measure 150V, 4.5A and 250 Watts. Determine the equivalent circuit parameters."

 I asked him to show me how those measurements were done. So he started to draw out the circuit model, which looks something like this:

Okay, I said, where do you measure the 220 volts? He started to point to one of the components, I don't remember which one, it might have been Xm, and I said: "No, you can't measure the internal parameters, those are only theoretical constructs. You can only measure the actual transformer".

He didn't exactly get it. So I drew this picture:

"THIS is what a transformer looks like", I told him. "There are only four wires. You have a voltmeter, an ammeter, and a wattmeter. Where do you hook them up?"

He was stumped. He had to admit that he had no idea.  They never talked about that in class. The prof told them that there was something called an "open circuit test" and something called a "short circuit test", that you get these measurements, and then you put them into these formulas, and the result is the equivalent circuit parameters (the ones you see all over the transformer model in the first diagram). No one ever talked about what it actually means.

And that's why I don't like models. Because they fool you into thinking that you know what you're doing, when you really don't. Actually, this particular student was pretty smart. He recognized right away that he'd been strung along, but he'd gone along with it because he had no choice. You follow directions or you fail. There's no time to second-guess the system and question what you're learning. And that makes him an exception.

The problem with education system is that by the time they've gotten this far, most students are no longer capable of recognizing what's wrong with the whole scenario. If I'd have confronted the typical engineering student with the fact that he was doing the calculation blindly even though he didn't even know where the voltmeter was supposed to be hooked up, he would have simply replied that it didn't matter, that you didn't need  to know that stuff because the right way to do it was just to follow the steps that the professor had laid out. And he would get the right answer on the test.

But that's not the biggest problem. The real tragedy of the education system is that it is doing exactly what society demands of it: churning out obedient workers for the government/industrial bureaucracy who will do what they are told without questioning or even trying to understand the reason behind it.

Why I Hate YIVO

We had an interesting discussion going about Galois Theory over the last little while, so I'm a bit loath to change direction in mid-stream, but I'm still working on getting all my Jewish Post articles up on the internet. So here's another one, this time about Yiddish spelling.

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My regular readers will have noticed that sometimes I include a short passage of transcribed Yiddish in my articles. Last week I quoted the memoirs of Yekhezkel Kotik, where he desccribes the jealousy of the poor Orthodox village priest at the luxurious lifestyle of his Polish Catholic counterpart, with his lavish mansion and his four beautiful “sisters” living together with him:

“Nur der orimer Rusisher galakh, velkher flegt platsn far kine fun dem raykhn lukses-lebn fun dem katoylishn galakh, hot far zayne poyerim, di poreytsishe layb-knekht geshvoyren, az di sheyne fraylayns zaynen gor nisht zayne shvester, zey zaynen im vild-fremde, kokhankes zaynen zey im…”

Except that’s not exactly the way it appeared last week. (The original is shown at the bottom of this article.) What you see above is the official transcription system mandated back in the 1920’s by the Vilna-based Jewish Scientific Institute, the  Jüdische Wissenschaftlicher Institute, or YIVO (the acronym being derived from their own phonetic spelling: Yidishe Visenshaftlikhe Institut). It was kind of like our own version of the French Academy, except for Yiddish.

Just as the French Academy sees one of its main goals as the safeguarding of the purity of the language from foreign (especially English) pollution, so was YIVO’s greatest concern in those days the incursion of Germanisms, which were known as daytshmerisms. (That word is a bit of a puzzle, by the way, for which I have never found a wholly satisfactory explantion. My best theory is that the ending comes from Mähren, the German name for Moravia, hence Deutsch-Mährisch.)

Expecially offensive to YIVO was the suggestion that Yiddish was not a real language, but merely a zhargon, a corrupt version of German. Thus motivated, we can see why YIVO would have sought to impose a stricty phonetic spelling system where, for example, “chutzpah” is spelled khutspe and “schmaltz” is spelled shmalts. In effect, YIVO proclaims to all the world that whatever Germanic or other origins a word may have is simply…irrelevant. I remember a conversation I once had with a Yiddish academic as to the origin of a word, I think it was golen, meaning “to shave”. Was it a German word or a Hebrew word, I asked? The Professor was unperturbed. “It is a Yiddish word”, he answered with finality, as though that were all that needed be said.

If there’s anything I’ve learned about Yiddish, it’s that the origins of a word were anything but “irrelevant”. We are taught in school that some English words come from Latin, and some come from Greek; but when we speak English, those origins are completely invisible to us. They’re all just words. Yiddish was totally different. Every Yiddish speaker, no matter how uneducated, knew instinctively if a word came from German, Hebrew, or Russian. The nuance carried by a word or phrase was often strongly influenced by its origins. The YIVO academics could fume that it was demeaning for writers to substitute loftier-sounding German expressions for everyday Hebrew terms (like gesicht instead of ponim, as Yehoash did in his translation of the first verse of Genesis); but like it or not, that kind of “code-switching” was deeply ingrained in the language and the culture. Often it was used for humorous effect. Either way, the mixed heritage of Yiddish was a defining aspect of its character, and not something to be ashamed of.      

But more than that, I think we’re cutting off our nose to spite our face if we ignore the German yikhus of our language. If we care at all about preserving Yiddish, then its relationship to German is for my money the biggest asset we have. We have a huge body of literature, historical writing, and music which we, in North America, have all but abandoned. We send our children to Hebrew School, and then to University where they can fritter away years taking courses in History of Film or Feminist Psychology or whatever, and I’m saying maybe they should take a course in Intro German. Because with a smattering of Hebrew and German under your belt, you’d be surprised how accessible that enormous body of Yiddish becomes, and what a window it opens into our past.

So my solution is that we transcribe Yiddish in a way that reflects as much as possible the two great languages from it is descended. There are some suttleties here and there involving vowel shifts which I think I’ve dealt with rather well with a cunning system õf döts ând squiggles. Maybe I’ll talk about it in more detail another time. But for now, here is the passage I started out with, re-written using my Germanized system. I think compared with the YIVO phonetics (at the top of this article) it looks pretty cool my way:

“Nur der ârimer Russischer galakh, welcher flegt platzen far kinah (envy) vun dem reichen luxus-leben vun dem Kathòlischen galakh, hât var seine pauerim, die poretzische leib-knecht (the squire’s serfs) geschwòren, as die schöene Fräuleins seinen gâr nischt seine schwester, séi seinen ihm wild-fremde (total strangers), kokhankes seinen séi ihm…”

Jewish Mathematics

The other day, following my introductory post on Arnold's proof of the fifth degree, one V. I. Kennedy posted a link to this remarkable lecture by a prof at the University of Toronto:

http://drorbn.net/dbnvp/AKT-140314.php

I didn't get a chance to look at the link until yesterday, because I was away. I wish Mr. Kennedy had given a little more information when he posted, because it turns out I was in Toronto  at the time and if I had known, I definitely would have called on the professor, one Dror Bar-Natan. I think we would have had an interesting conversation.

What makes me think that Dror and I would have anything in common? Well, for one thing both of us had the same reaction to this video by Boaz Katz: namely, both of us, as soon as we saw the video, had to drop whatever else we were doing and tell the world about this fantastic "new" proof of fifth degree ("new" in the sense that it was just in the 1960's that Arnold came up with it.)

But that's not all. When Dror tells his class why he felt driven to talk about this proof, he explained it in almost the same words I used twenty years ago when I was the math guy on community access TV here in Winnipeg: how, when he was a student, he took a whole course in abstract algebra, sitting through a labyrinth of theorems, lemmas, and corroloraies, until one day near the end of the course the professor announced "...and therefore the fifth degree is unsolvable"; and how it had always bothered him that for all that effort, he still never really understood why. If you watched Dror's video, you might check out this old Math with Marty episode where I say the same thing.

The other funny thing about Dror's lecture is he goes off on a little tangent about how you can understand that the square root of two is irrational. It's funny that the "alternate" proof he shows is almost the same as the one I did on this Math with Marty episode, except Dror refines it down to the bare bones while I work it through using numerical examples. Dror also has a hilarious punch line where he shows a "simple" one-line proof of the irrationality of any higher-order nth root of two: the joke is that his proof relies on the fact that a rational nth root of two would be contradicted by Fermat's Last Theorem, whose "proof" of course takes about nine hundred pages.

But beyond all that there's one more thing that you just can't avoid noticing through all this: it's the Jewish presence. All of us here in this discussion are Jews. Dror is a Jew. Boaz Katz is a Jew. I'm a Jew. Even V.I. Arnold is a Jew. Yes, Balarka Sen is a thirteen-year-old kid from West Bengal, but he's the exception that proves the rule. What's going on here?

Yes, we know the Jews are smart. But there's something more going on. We're brought up to believe we're smarter than everyone else, and maybe we are just a little, but it turns out there's a lot of smart goyim out there too. And by the way, if we believe that we're so smart, we also have to be prepared to accept that we have other distinguishing characteristics, which aren't always of the positive persuasion, so to speak. I wrote about this a couple of years ago in this blogpost, "Jewish Lightning", which asks the question: do Jews burn down their stores to collect the insurance money? But I digress.

Quite apart from the hypothetical question of how smart the Jews are, I think the present discussion illustrates another reason why Jews are high achievers in math and physics. I think we have a different aesthetic sense of math than your average white person. Dror is clearly excited about Arnold's proof. Boaz Katz is excited about it. If you look up physics lectures by people like Feynaman or Walter Lewin of MIT, they are clearly excited by their topics.

But it's more than just that. What we (the participants in the present discussion) all have in common is that we are excited by the beauty and elegance of an explanation which allows us to understand these things in human terms. I'm not saying that your average white person isn't also capable of experiencing the same emotional response to a math proof. It just seems to me that they aren't driven that way to the same extent. It's like you can teach a dog to walk on its hind legs, but it isn't exactly natural.

I've met enough regular white people who are "smarter" than me in my travels as a university student to be fully aware of my own limitations and the potential abilities of others. But when I compare the very smart people who I've known to the Jews like Boaz and Dror (also smarter than me), not to mention Feynmann and Lewin, I think I can pinpoint a qualitiative distinction. Most of the really smart white people I've met in math and physics seem to have an ability, incomprehensible to me, to be able to read a mathematical proof the way you or I read a newspaper article. I don't begin to know how that is possible. I can only "read" a proof line by line if I already figured out in my head what it means and where it's going.

I'm not saying some Jews don't have that purely analytical ability as well. Obviously Feynmann did. I'm just saying that I'm able to function at a pretty high level in math and physics working on a purely intuitive level. The conclusion I'm drawing is that the over-representation of the Jews at the very highest levels isn't necessarily because we're smarter than regular white people, but because we produce individuals who combine a high analytical ability with an intuitive approach driven by out unusual aesthetic sense: the same aesthetic sense that produces a Barbara Streisand or a Leonard Cohen.

In the old country, we even had a word for that quality: we called it the pintele Yid, the Jewish spark or the "point of light". I don't think we even knew exactly what we meant by the expression, but we knew there was something there that needed its own word.

EDIT: I sent Dror an email to tell him about the discussion we've been having here, and he wrote back the next day to tell me he'd read my blogpost, but he had a small correction to make: namely, he says he's not Jewish. I guess it's possible, but I still think he might be just a little bit Jewish. In fact, I think he looks quite a lot like the guy in The Princess Bride. What do you think?

POST-SCRIPT: I've been going over the video proofs and I think both Boaz and Dror have left out something important. I still think Arnold's proof is valid, but I don't think either of these guys have presented it correctly. I'm going to email them and ask them to respond to my criticism and we'll see what they say.

Here is my problem: I've watched both their videos again, and if I follow their logic exactly, it seems they've both proven that you can't solve the basic quintic equation:

x^5 - 2 = 0

Look at the roots. There are five of them. You can easily generate closed-loop excursions of the coefficients of the above equations in the complex plane which have the effect of reshuffling the five roots in any arbitrary order. We then look at arbitrary combinations of commutators of these permutations, and we find that it is possible to construct infinitely high towers of commutators that fail to return the roots to their original order. But we can also prove that if the roots are expressible as complicated expressions of nested radicals ("solvable in radicals"), that any sufficiently high tower of commutators must return the roots to their original order. This is a contradiction, and therefore the equation is not solvable in radicals.

Am I wrong, or do Boaz and Dror both fail to explain this case in their videos? Yes, it's a mistake that can be corrected, but I don't think it's a trivial mistake. Because to fix this mistake I think you have to invoke some Galois theory, and it seems that avoiding Galois theory was one of the big "selling points" of Arnold's method in the first place.

Someone tell me I've got this wrong...

V. I. Arnold's Topological Proof

I just came across the strangest thing. This Israeli guy put up a youtube video about how you can prove the fifth-degree equation is unsolvable, based on the ideas of a Russian mathematician named V. I. Arnold. I haven't worked it all out yet, but it starts off with the strangest idea. You take an equation, and then plot out the roots in the complex plane. Then you also plot out the coefficients  in the complex plane. Why the complex plane, you ask? The cooefficients are all integers. Why not just plot them on the number line? Answer: because we're about to mess with them.

This is the funny thing. Take the simplest possible equation, like:

x^2 - 2 = 0.

The coefficients are 1 and 2, and the roots are +/- sqrt(2). Here is what we are going to do. We are going to take the "2" (the constant coefficient) from the equation and move it slowly along a circle about the origin. And we're going to observe what happens to the solutions of the equation as it changes.


Try it yourself! You'll see that when you complete the circle, so that the equation returns to its original form, that the roots of the equation also go back to their original values. But they are reversed! You have to go around the circle twice to put the roots back where they started. It's the strangest thing.

In the video, the Israeli guy claims (without really explaining it all that clearly) that in general, you can construct loops in the map of the coefficient such that by dragging the coefficients around those loops, you can arbitrarily force every possible permutation of the roots. I've shown you what happens when you drag the "2" about a loop in the complex plane - in the map of the roots, the two square roots of two switch places. Arnold's idea is that in general, you can force every permutation of the roots by dragging the coefficients around the complex plane.

I haven't yet figured out why this must be so (EDIT: Okay, I've thought about it and it's true: Boaz explains it around 4 minutes into his video), but if  since it's true then it has consequences. The idea is that if there is a solution to the fifth degree, it has to be written in terms of the coefficients arranged somehow within a complicated nested system of radicals...but for any such representation, there are restrictions as to where the roots can go when you mess with the coefficients.

How does this help us? Well, for one thing, in the example I've just shown, it proves that the solutions of x^2 - 2 cannot be rational. (EDIT: No, that's not quite right: it only shows that you need to write them with a formula that includes a square root sign). Why? Because the loop we constructed flips them around. But it's not so hard to see that if the roots are given by rational expressions, then any loop of the coefficients in the complex plane has to bring each of those rational expressions (for the roots) right back to where it started. So rational expressions don't flip around with each other, the way we did with the square roots of two.

Anyhow, that's the basis of Arnold's proof, which I'm not able to go much farther into right now. But it's something to think about.

If you're a follower of my blog, you know I've written a lot about the fifth degree equation. I think I explain it pretty well here at Why You Can't Solve The Qunitic. But I've never seen anything like Arnold's method before.,

How Light and Sound are Different

The other day I started doing the Doppler shift as a relativity problem. It starts off looking a lot like the Doppler shift for ordinary sound in air. I had Alice sending a series of sound/light pulses to a train travelling at 4/5 the speed of sound/light; and I had Bob on the back of the train with a mirror reflecting the sound/light back to Alice. It's not too hard to calculate that if Alices pulses are 1 microsecond apart, then when they come back to Alice they are 9 microseconds apart, for a Doppler shift of 900%. You can see it easily from the picture below. The calculation is correct for light, and it's correct for sound:

What gets funny if we asked how much Doppler shift Bob sees. It's not hard to do the calucation for sound in air. We just drop vertical lines from Bob's pulse detection points down to the time axis. It's not hard to see that Bob detects the pulses 5 microseconds apart, for a Doppler shift of 500%:

Here's the thing: Bob is sending out pulses 5 microseconds apart, and Alice is measuring them 9 microseconds apart. So Alice's Doppler Shift is 180%. That's not the same as Bob's. So by analyzing who has a greater doppler shift, they can figure out that Alice is stationary and Bob must be moving.

And that's not how relativity works. In relativity we're not allowed to distinguish the stationary from the moving observer, so both Alice and Bob have to measure the same doppler shift. It's almost impossible to see how they can do that...unless we realize that time is moving slower for Bob than for Alice.

The total doppler for the reflected pulses is 900%. And the only way to make it the same for both observers is for them both to see 300%. Bob sees the pulses 3 microseconds apart, and Alice sees them 9 microseconds apart.

We did the caluclation for the special case of the train going at 4/5 the speed of light. But if we let the speed vary from 0 to 100%, and trace the contour defined by the equal-interval ticks, we will find that they are the hyperbolas defined by the equation x^2 - t^2 = constant: