10 ms·

New straighforward approach to teaching quantum mechanics

155 points by georgecmu 14y ago

geuis 14y ago
By completely random occurrence, I was watching this video yesterday: http://www.youtube.com/watch?v=_Cv5ldhxpLA&feature=g-hist http://www.youtube.com/watch?v=_Cv5ldhxpLA&feature=g-his... Its by the fellow that does Minute Physics on Youtube. Towards the end of his talk, he specifically talks about this topic. He makes a comparison between mathematics and physics, wherein mathematics you are taught the math without a ton of history. No scaffolding. In physics, you are taught the entire history of physics, i.e. scaffolding, from the ground up. He also advocates directly teaching modern theory and leaving most of the history out. Make learning physics more like learning math.
- nhaehnle 14y ago
  As a PhD student in mathematics, I have to say that history is valuable in mathematics because it explains why people are interested in certain problems. When I was a master level student, I always got annoyed by advanced courses in algebra (things like homological algebra) where it was often unclear why one should be interested in the problems stated there. Prompting the profs to give some short historical overview can be very enlightening. I do believe you can leave the vast majority of the history out. Advances in notation etc. were made for a reason. But a little bit of history can be quite important for context.
  - derleth 14y ago
    > I have to say that history is valuable in mathematics because it explains why people are interested in certain problems. Motivation is essential, but there are motivations other than the ones that historically lead to the creation of the field. In number theory, one of my favorite examples, the motivations that lead people to be interested in it now come from modern cryptography, which flatly didn't exist when the field was founded but provides more interesting and relevant examples than the obsessions of century-old mathematicians.
  - btilly 14y ago
    History may be valuable in mathematics, but it is demonstrably skipped. Few know much about it. When I was in grad school in math, I made the interesting discovery that if p and q are polynomials over a commutative ring, any polynomial that is symmetric in the roots of p and q is actually a polynomial in the original ring in the coefficients of p and q. (The construction works whether or not the ring can be embedded in a field where said roots actually exist.) Using this observation it is trivial, for instance, to write down in fully factored form a polynomial that has sqrt(2) + cube_root(3) as a root. This construction was news to various mathematicians that I talked to, including a combinatorics prof who studied symmetric polynomials and a number theorist who worked on stuff related to the algebraic integers. Then finally I talked to a very old mathematician with an interest in history. He told me that I had rediscovered an old way to do things. At his encouragement I went to the library, and picked up an algebra book from the 1800s. My construction was taught, and there was a whole chapter full of problems where students were expected to use it to come up with polynomials with specific roots. Furthermore as I looked into it both of the professors that I mentioned before worked in areas whose history dated back to the observation that I mentioned. It was used in the original proof that the algebraic integers form a ring, and that construction was the original reason that people were interested in symmetric polynomials. For another demonstration of how little of their own history mathematicians know, ask anyone why the notation for the second derivative is d^2y/dx^2. Then ask them where the f' notation comes from. Then ask them what Cauchy was trying to do that lead to Cauchy sequences. Most will draw a blank on all three. Don't read on until you're satisfied that you don't know the answers. In the original infinitesmal notation, d was an operator. It could be defined by d(y) = y(x + dx) - y(x). And you'd calculate a slope as dy/dx (drop any infinitesmal bits). Well when you work out d(dy/dx)/dx it turns out that you get d(d(y))/(dx * dx) which is more compactly written d^2y/dx^2. The f' notation was introduced by Lagrange in an attempt to get rid of infinitesmals by defining differentiation as a formal algebraic operation on polynomials and power series. This fell apart when Fourier demonstrated that apparently well-behaved power series could be used to construct pathological things like step functions. Cauchy came up with Cauchy sequences while attempting to define infinitesmals rigorously. His approach fell apart on the seemingly trivial example of how you rigorously prove the chain rule when the derivative of the inner thing is 0. (He was trying to avoid 0/0, but in that special case you get 0/0 all over the place.)
- sp332 14y ago
  In chemistry class (where we learned particle physics in high school) I really hated having to unlearn all the crap we were quizzed on in the last section to learn some new crap that I'd be tested on and then expected NOT to use later because it was obsolete! /rant Anyway, if you really want to teach the history of a subject like math, you should probably do it after teaching the modern understanding of the subject. In math class our teacher explained some of the controversy over the invention of calculus, and the origin of the different notations. It gave us some appreciation for the unintuitive and tricky nature of the subject, which looks very simple now.
  - derleth 14y ago
    > the controversy over the invention of calculus ... which lead to the less-intuitive epsilon-delta proof framework replacing the much more intuitive infinitesimal framework calculus was originally founded on, until the 1960s when infinitesimals were reformulated as part of nonstandard analysis. (Unless the controversy you're talking about is the utterly uninteresting one about Leibniz vs Newton.) Another example where a historical development would lead students through a completely pointless diversion (epsilon-delta proofs) simply because Robinson was born in 1918 instead of 1618.
    - sp332 14y ago
      I did mean Leibniz vs. Newton but I remember infinitesimals being glossed and wishing we would get into them more.
    - SatvikBeri 14y ago
      I was trying to play devil's advocate but I honestly couldn't think of a good reason to teach Calculus using epsilon-delta vs. Nonstandard Analysis. The way Calculus is taught today is already unrigorous until you get to Analysis, so there's no strong reason to teach students using epsilon-delta first.
      - jasomill 14y ago
        Right, because students struggling with "for all P there exists Q" won't have any problem with higher-order logic? It's always seemed to me a more sensible road to both simpler and more rigorous calculus courses would somehow involve reducing the scope of functions under consideration, since most of the exercises involve analytical functions anyway. Then, when students are ready for analysis, it can be more about "how to reduce nasty cases to problems you know how to solve, and how to recognize the truly pathological specimens where you can't" rather than "everything you thought you knew is wrong."
- beagle3 14y ago
  That's not true for a lot of subjects. For example, every math program I've looked at teaches integration of complex functions the way it was arrived at historically, which involves increasingly complex shapes in the plane that eventually lead to the general case - some program spend half a semester going through this construction. However, if you're familiar with vector calculus, it is an immediate result of Green's theorem in the plane - doesn't even take 5 minutes of work. Engineering programs often go that route. And if you structured the material in the "right" way, it would be even simpler - Green's theorem in the plane is itself a special case of Stokes' theorem; However, "simpler" is relative - stokes theorem is much more abstract. So it's "simpler" in the sense that you can prove much, much less. It's more complex in the sense that you have to grok all those higher-level abstractions without a mental picture. Sort of like software engineering - LISP is a better fundamental system. But it is too abstract for most people in the field. And so are APL/J/K, although in a different way.
- alan_cx 14y ago
  No, no, no, no. Instinctively I recoil at that idea. I dont quite know how to explain why, but I'll give it a go... Im good with computers because I was there in the early days when it was pretty simple. Everything since has build upon those simple blocks. So, I can always work it back as it were. It kinda means I can simplify current complexities down to simple fundamentals. So, it leaves me with an ability to be presented something new, or different and very quickly I can understand it. Does that make any sense? I assume many people here are exactly like that, or recognise what Im trying to say. Pathetic I know, but if any one can put it better, please do!!!! Anyway, all I know is that I am so much better off knowing how it all came together than people who don't. I don't understand how any one can claim to understand any subject with out understanding it's journey as it were. I make a distinction here between know how to use something and understand it. If all one wants to do is "use", then understanding isn't absolutely necessary. You know, plenty of people drive cars with out understanding anything about how they work.
  - SatvikBeri 14y ago
    No one is saying that knowledge of the history isn't useful. But it's not the most efficient way to learn the concepts. For example, it would be crazy to start learning basic programming by first studying the physics behind computers. Learn the history behind your subject, but separate it from the pedagogy. Otherwise you'll have a much harder time learning knowledge that you can actually apply.
    - derleth 14y ago
      > For example, it would be crazy to start learning basic programming by first studying the physics behind computers. Technically, a truly historical development in terms of teaching programming would involve plugboards and specialized hardware design freshman year, punch cards sophomore year, teletypes junior year, and then 'intelligent' terminals senior year. We simply wouldn't have time to teach anything similar to modern networking, which only really came about in the 1980s, let alone GUI design or Web programming.
  - derleth 14y ago
    > I was there in the early days when it was pretty simple. I think you have an interesting idea of 'simple': A lot of software becomes orders of magnitude simpler to design and reason about once you have enough RAM to organize it in the obvious way. For example, for many parsing tasks, a recursive-descent parser is the obvious way, but you can only design your software that way once you have a language that can directly express recursion and a computer that can grow a big enough call stack (on the stack or in the heap) to allow the parser to juggle nontrivial sentences. Otherwise, you're left the laborious, non-obvious, labor-intensive, but ultimately pointless task of turning your design into a program your tools can implement. > So, it leaves me with an ability to be presented something new, or different and very quickly I can understand it. Honestly, I think that says a lot more about your intelligence than your history. That's actually a pretty fair definition of intelligence, in fact. How about this: Teaching networking now is teaching TCP/IP and everything that TCP/IP rests upon, such as Ethernet and Wi-Fi. Theory naturally flows from practice, and that is the practice. The OSI Model? Gone. Out. Forget it. Nobody actually implements all seven layers; at most, we have four. NCP? What? If you know what NCP stands for, congratulations, you know something entirely useless. "Jeopardy!" would love you. So why teach the stuff we know is useless? It actively discourages students. It makes them think they're just wasting their time, primarily because they are. The history of computer networking is a valid topic. It should be taught in its own course, it deserves its own course. It does not deserve to be shoved into a course on how networking actually works.
glimcat 14y ago
I'm sold, although I'd still start by sitting them down with a copy of Feynman's QED. When doing math about a thing, it helps tremendously if you already have some sort of intuition about what the math is describing. Also, the historical stuff is fun and interesting. Physics students should probably be taught this stuff regardless, but maybe not as part of a class on theory.
- jasomill 14y ago
  On the other hand, Feynman's most important lesson is thoroughly modern: the surest way to understand something "intuitively" is to create it.
  - derleth 14y ago
    > the surest way to understand something "intuitively" is to create it. And there's no reason the best way to create something has to be the way it was created the first time.
  - Tichy 14y ago
    Cool idea. Screw learning QM - let's rediscover it on our own!
pirateking 14y ago
I find historical narrative to be an effective way to learn, but just reading names, dates, and important events is boring. You have to pretend like you have a time machine. An effective historical lesson takes you back in time and puts you right at the scene of the development, almost allowing you to have a conversation with the characters and follow along with their thought process. That being said, I appreciate this straightforward approach just as much, and find that having a good understanding of the core concepts first, makes adjusting the settings on the "time machine" much smoother.
- derleth 14y ago
  > allowing you to have a conversation with the characters and follow along with their thought process. Which were muddled, confused, and, pretty much by definition, partially incorrect. Historical development necessarily involves recapitulating the confusions of the people who finally got confused enough to throw over their original ideas. Even if you trim the history down to only the path that actually lead to modern theory, you still have a lot of confusion to get through before you can actually introduce the main idea of the topic in all its glory. And that's in addition to the confusion students have naturally.
  - pirateking 14y ago
    Very true. Though there is still some value in the journey through history, aside from learning the main idea. It can serve as inspiration for your own quest, and also challenge you to question modern theory as those who came before did. Being equipped with some fundamentals first, makes exploring the history less confusing. I think Charles Petzold's book "Code", is a great example of blending historical narrative with core concepts - I highly recommend it.
- Retric 14y ago
  IMO, the simple approach is dangerous. Saying here is the theory have at it is fine if you want to teach someone how to design transistors, but the point is not that QM is correct it's just the best approximation yet and in no way the 'holy word' on how the universe operates. The single most important thing to teach science students is to discard existing theory's when reality does not support them and to require any new theory to also be supported by the body of existing experimental evidence.
karpathy 14y ago
I took 2 courses on Quantum Mechanics during my undergrad and I'm taking the Coursera course now as well for fun. My biggest quibble with how QM is presented is that it is, paradoxically, never tied into physics. Quantum Physics is really a Linear Algebra + Statistics but with extension to complex numbers. In my experience it is just another math course and there is no Physics anywhere. There's always talk of measuring states, applying Hadamard gates and writing down their decomposition in the eigenbasis but it's abstract and meaningless. What does a particle with some particular wave function look like? As it evolves in time it "smears" out according to the time evolution equations, but how fast is the process? Does it occur on scale of nanoseconds? Seconds? Hours? How is it suspended, or operated on? What exactly is an example of a measurement? There are so many tangible questions that would help with intuition, but are never addressed. This would be my approach to teaching quantum mechanics. Connect it to a concrete physical system, explore in detail going back and forth between experiment and math. And best, maybe even simulate the system somehow. Some time ago I made an effort to try to simulate what I learned and this was the result (as an example): http://www.youtube.com/watch?v=a88GlrUmI9Y&feature=plcp http://www.youtube.com/watch?v=a88GlrUmI9Y&feature=plcp I'ts ugly and it's probably wrong, but it's tangible and the best I could do because finding this kind of Quantum Mechanics, as opposed to a lot of talk about measuring things is very hard.
- andrewcooke 14y ago
  yeah, but this is more general than qm - it's amusing to ask (other) physicists how fast electrons move in typical wires (a purely classical number, but most won't have a clue).
  - jules 14y ago
    It's not exactly clear what you mean by "velocity" in that question. Drift velocity? Fermi velocity? (and others) These two differ by multiple orders of magnitude, so it's easy to wrongly conclude that another physicist doesn't have a clue. Intuitive explanation: drift velocity is the overall streaming velocity of the moving electrons, fermi velocity is the velocity of a single electron. The reason these are different is that electrons move back and forth heavily, thus the fermi velocity is high. But under normal voltages, they move just slighly more in one direction than the other on average, which is the drift velocity. Of course this is highly simplified, and the real story is quantum mechanical.
gus_massa 14y ago
Perhaps this is a good approach to Quantum Mechanics for a course about Quantum Computers. (Maybe is has too much emphasis is the probability part.) Quantum Mechanics has a little set of clear rules that are easy to follow and get the correct result. So if your plan is to have some little black quantum boxes and combine them to get quantum computations perhaps it is a good idea to jump directly to the rules and forget about the history part. The problem arises when you really want to understand how the little black quantum boxes work, or how this scales to bigger systems and to the real work. The correspondence principle says that if the system is big enough the quantum effects should almost disappear, and the system should be correctly approximated by Newtonian Physic. (There are some macroscopic effects that depend on the quantum nature of the world, the most clear is superfluity.) The biggest problem is the correspondence, how to make the connection between the real world that you see and the quantum physics. The pseudo-historic path gives in each step a more complex model, but the jumps are smaller. (For example, usually the Schrödinger's model is explained before the Heisenberg's model because it is more intuitive, but Heisenberg's work was a few years before Schrödinger's work.) In this approach, there is no reference to the method that you should use to transform a classical model to a quantum model. The main examples are the harmonic oscillator and the hydrogen-like atom that are used as the base to explain the properties of the matter.
raylu 14y ago
This sounds a lot like what I started reading at LessWrong: http://lesswrong.com/lw/r5/the_quantum_physics_sequence/ http://lesswrong.com/lw/r5/the_quantum_physics_sequence/ One line that particularly stuck with me is: "Dragging a modern-day student through all this may be a historically realistic approach to the subject matter, but it also ensures the historically realistic outcome of total bewilderment. Talking to aspiring young physicists about 'wave/particle duality' is like starting chemistry students on the Four Elements."
- gus_massa 14y ago
  In Physics sometimes and approximated theory / model is useful, in spite it is incorrect. The problem is that the Schrödinger / Heisenberg quantum states of a particle are a lie. If you have an electron, it doesn't follow the Schrödinger / Heisenberg equation. It can emit a photon an reabsorb it a little time after. This is not part of that equation and has a very easy to measure effect that is the Lambs Shift http://en.wikipedia.org/wiki/Lamb_shift#Lamb_shift_in_the_hydrogen_spectrum http://en.wikipedia.org/wiki/Lamb_shift#Lamb_shift_in_the_hy... . And the electron can even do more crazy things, but they are luckily more difficult to measure. This is the reason to use Quantum field theory. * If you are only going to buy a lens to put in front of your photodetector, then probably the wave/particle duality is a good enough approximation (in spite that it doesn't make sense). * If you are doing quantum-chemistry then use the Schrödinger equation (in spite that it doesn't work for strong electric fields and is not compatible with special relativity). * If you work near a big particle accelerator you should try at lest to use the standard model (in spite that renormalization doesn't make sense). * And I hope that you never have to use string theory to explain and experiment.
- thebooktocome 14y ago
  It's nothing like the quantum physics sequence. Yudkowsky starts out making up numbers from nowhere and asserts things for purely philosophical reasons, whereas Aaronson actually has proofs (albeit left to the reader).
  - loup-vaillant 14y ago
    Sure, the numbers in "Quantum Explanations" are made up, but (1) the experiments are real, and (2) everything besides the numbers is accurate (as far as I know). Plus, the goal of the sequence was never to actually explain quantum physics. It was to explain why a realist perspective (the wave function is all there is, and the math says it doesn't collapse, so it really doesn't) is by far the most probably correct, despite the fact that it makes no new prediction compared to previous interpretations. That, plus answering some philosophical question with physics. Aaronson's explanation definitely is a step in the right direction. I still have a quibble however: he keeps mentioning "probability" as an analogy to amplitude. That confuses his explanation in my opinion. I'd rather have a straight explanation of QM math, then an explanation about its similarities with probability theory. And the mixed state paragraph seems to conflate subjective probability and actual distribution of amplitude. Ick.
gwern 14y ago
> Two other perfect examples of "obvious-in-retrospect" theories are evolution and special relativity. Admittedly, I don't know if the ancient Greeks, sitting around in their togas, could have figured out that these theories were true. But certainly -- certainly! -- they could've figured out that they were possibly true: that they're powerful principles that would've at least been on God's whiteboard when She was brainstorming the world. Actually, people studying the Pre-socratics like to point out that Anaxagoras (IIRC) theorizes something eerily close to evolution: that at the beginning, there were all sorts of random creatures, and only the ones which did well survived and created more creatures like them. I don't know why noone followed up on it (in contrast to something like heliocentrism, where we know why the Greeks abandoned it for what were excellent and unobjectionable reasons at the time).
bobwaycott 14y ago
I find the references to "God" (e.g., "...why did God choose to do it that way and not some other way?") to be both distracting and unhelpful working through the text itself. It signals to my brain (possibly incorrectly) that either (1) the author is trying to inject an unwarranted religious idea into an otherwise potentially helpful explanation in a devious way, or (2) the author has sadly mis-chosen a loaded term that doesn't add anything helpful to the explanation at all, and instead detracts from working through it because it creates the nagging question in my head of, "Is he really suggesting that God (whatever that may be in the reader's mind) chose to do this?". A better option (for just the chosen example) would be: "... why does the Universe do it that way and not some other way?".
- thebooktocome 14y ago
  This is exceedingly common language in the mathematical community, particularly for the generation of Scott Aaronson and the one before him.
  - bobwaycott 14y ago
    Oh, okay. Thank you so much for pointing that out to me. I somewhat assumed as much, and I honestly wasn't intending to create anything remotely ad hominem in my critique. Since it's not so common in hardly anything I typically read, it just stuck out strangely (I am only just digging into understanding QM/QP). Again, sincere thanks.
  - andyjohnson0 14y ago
    Are they referring to a God in the religious sense, or using it as a shorthand for "nature" or "the universe"?
    - thebooktocome 14y ago
      Some of them are no doubt serious; I seem to remember a study showing that mathematicians are more likely to be theist than atheist. In my experience, "Why did God pick X?" and the like is shorthand for, "Is there a classification/uniqueness theorem that says X is the only possible solution to the problem?", which is how it is used here.

155 points by georgecmu 14y ago

geuis 14y ago

By completely random occurrence, I was watching this video yesterday: http://www.youtube.com/watch?v=_Cv5ldhxpLA&feature=g-hist http://www.youtube.com/watch?v=_Cv5ldhxpLA&feature=g-his... Its by the fellow that does Minute Physics on Youtube. Towards the end of his talk, he specifically talks about this topic. He makes a comparison between mathematics and physics, wherein mathematics you are taught the math without a ton of history. No scaffolding. In physics, you are taught the entire history of physics, i.e. scaffolding, from the ground up. He also advocates directly teaching modern theory and leaving most of the history out. Make learning physics more like learning math.

nhaehnle 14y ago
As a PhD student in mathematics, I have to say that history is valuable in mathematics because it explains why people are interested in certain problems. When I was a master level student, I always got annoyed by advanced courses in algebra (things like homological algebra) where it was often unclear why one should be interested in the problems stated there. Prompting the profs to give some short historical overview can be very enlightening. I do believe you can leave the vast majority of the history out. Advances in notation etc. were made for a reason. But a little bit of history can be quite important for context.
- derleth 14y ago
  > I have to say that history is valuable in mathematics because it explains why people are interested in certain problems. Motivation is essential, but there are motivations other than the ones that historically lead to the creation of the field. In number theory, one of my favorite examples, the motivations that lead people to be interested in it now come from modern cryptography, which flatly didn't exist when the field was founded but provides more interesting and relevant examples than the obsessions of century-old mathematicians.
- btilly 14y ago
  History may be valuable in mathematics, but it is demonstrably skipped. Few know much about it. When I was in grad school in math, I made the interesting discovery that if p and q are polynomials over a commutative ring, any polynomial that is symmetric in the roots of p and q is actually a polynomial in the original ring in the coefficients of p and q. (The construction works whether or not the ring can be embedded in a field where said roots actually exist.) Using this observation it is trivial, for instance, to write down in fully factored form a polynomial that has sqrt(2) + cube_root(3) as a root. This construction was news to various mathematicians that I talked to, including a combinatorics prof who studied symmetric polynomials and a number theorist who worked on stuff related to the algebraic integers. Then finally I talked to a very old mathematician with an interest in history. He told me that I had rediscovered an old way to do things. At his encouragement I went to the library, and picked up an algebra book from the 1800s. My construction was taught, and there was a whole chapter full of problems where students were expected to use it to come up with polynomials with specific roots. Furthermore as I looked into it both of the professors that I mentioned before worked in areas whose history dated back to the observation that I mentioned. It was used in the original proof that the algebraic integers form a ring, and that construction was the original reason that people were interested in symmetric polynomials. For another demonstration of how little of their own history mathematicians know, ask anyone why the notation for the second derivative is d^2y/dx^2. Then ask them where the f' notation comes from. Then ask them what Cauchy was trying to do that lead to Cauchy sequences. Most will draw a blank on all three. Don't read on until you're satisfied that you don't know the answers. In the original infinitesmal notation, d was an operator. It could be defined by d(y) = y(x + dx) - y(x). And you'd calculate a slope as dy/dx (drop any infinitesmal bits). Well when you work out d(dy/dx)/dx it turns out that you get d(d(y))/(dx * dx) which is more compactly written d^2y/dx^2. The f' notation was introduced by Lagrange in an attempt to get rid of infinitesmals by defining differentiation as a formal algebraic operation on polynomials and power series. This fell apart when Fourier demonstrated that apparently well-behaved power series could be used to construct pathological things like step functions. Cauchy came up with Cauchy sequences while attempting to define infinitesmals rigorously. His approach fell apart on the seemingly trivial example of how you rigorously prove the chain rule when the derivative of the inner thing is 0. (He was trying to avoid 0/0, but in that special case you get 0/0 all over the place.)
sp332 14y ago
In chemistry class (where we learned particle physics in high school) I really hated having to unlearn all the crap we were quizzed on in the last section to learn some new crap that I'd be tested on and then expected NOT to use later because it was obsolete! /rant Anyway, if you really want to teach the history of a subject like math, you should probably do it after teaching the modern understanding of the subject. In math class our teacher explained some of the controversy over the invention of calculus, and the origin of the different notations. It gave us some appreciation for the unintuitive and tricky nature of the subject, which looks very simple now.
- derleth 14y ago
  > the controversy over the invention of calculus ... which lead to the less-intuitive epsilon-delta proof framework replacing the much more intuitive infinitesimal framework calculus was originally founded on, until the 1960s when infinitesimals were reformulated as part of nonstandard analysis. (Unless the controversy you're talking about is the utterly uninteresting one about Leibniz vs Newton.) Another example where a historical development would lead students through a completely pointless diversion (epsilon-delta proofs) simply because Robinson was born in 1918 instead of 1618.
  - sp332 14y ago
    I did mean Leibniz vs. Newton but I remember infinitesimals being glossed and wishing we would get into them more.
  - SatvikBeri 14y ago
    I was trying to play devil's advocate but I honestly couldn't think of a good reason to teach Calculus using epsilon-delta vs. Nonstandard Analysis. The way Calculus is taught today is already unrigorous until you get to Analysis, so there's no strong reason to teach students using epsilon-delta first.
    - jasomill 14y ago
      Right, because students struggling with "for all P there exists Q" won't have any problem with higher-order logic? It's always seemed to me a more sensible road to both simpler and more rigorous calculus courses would somehow involve reducing the scope of functions under consideration, since most of the exercises involve analytical functions anyway. Then, when students are ready for analysis, it can be more about "how to reduce nasty cases to problems you know how to solve, and how to recognize the truly pathological specimens where you can't" rather than "everything you thought you knew is wrong."
beagle3 14y ago
That's not true for a lot of subjects. For example, every math program I've looked at teaches integration of complex functions the way it was arrived at historically, which involves increasingly complex shapes in the plane that eventually lead to the general case - some program spend half a semester going through this construction. However, if you're familiar with vector calculus, it is an immediate result of Green's theorem in the plane - doesn't even take 5 minutes of work. Engineering programs often go that route. And if you structured the material in the "right" way, it would be even simpler - Green's theorem in the plane is itself a special case of Stokes' theorem; However, "simpler" is relative - stokes theorem is much more abstract. So it's "simpler" in the sense that you can prove much, much less. It's more complex in the sense that you have to grok all those higher-level abstractions without a mental picture. Sort of like software engineering - LISP is a better fundamental system. But it is too abstract for most people in the field. And so are APL/J/K, although in a different way.
alan_cx 14y ago
No, no, no, no. Instinctively I recoil at that idea. I dont quite know how to explain why, but I'll give it a go... Im good with computers because I was there in the early days when it was pretty simple. Everything since has build upon those simple blocks. So, I can always work it back as it were. It kinda means I can simplify current complexities down to simple fundamentals. So, it leaves me with an ability to be presented something new, or different and very quickly I can understand it. Does that make any sense? I assume many people here are exactly like that, or recognise what Im trying to say. Pathetic I know, but if any one can put it better, please do!!!! Anyway, all I know is that I am so much better off knowing how it all came together than people who don't. I don't understand how any one can claim to understand any subject with out understanding it's journey as it were. I make a distinction here between know how to use something and understand it. If all one wants to do is "use", then understanding isn't absolutely necessary. You know, plenty of people drive cars with out understanding anything about how they work.
- SatvikBeri 14y ago
  No one is saying that knowledge of the history isn't useful. But it's not the most efficient way to learn the concepts. For example, it would be crazy to start learning basic programming by first studying the physics behind computers. Learn the history behind your subject, but separate it from the pedagogy. Otherwise you'll have a much harder time learning knowledge that you can actually apply.
  - derleth 14y ago
    > For example, it would be crazy to start learning basic programming by first studying the physics behind computers. Technically, a truly historical development in terms of teaching programming would involve plugboards and specialized hardware design freshman year, punch cards sophomore year, teletypes junior year, and then 'intelligent' terminals senior year. We simply wouldn't have time to teach anything similar to modern networking, which only really came about in the 1980s, let alone GUI design or Web programming.
- derleth 14y ago
  > I was there in the early days when it was pretty simple. I think you have an interesting idea of 'simple': A lot of software becomes orders of magnitude simpler to design and reason about once you have enough RAM to organize it in the obvious way. For example, for many parsing tasks, a recursive-descent parser is the obvious way, but you can only design your software that way once you have a language that can directly express recursion and a computer that can grow a big enough call stack (on the stack or in the heap) to allow the parser to juggle nontrivial sentences. Otherwise, you're left the laborious, non-obvious, labor-intensive, but ultimately pointless task of turning your design into a program your tools can implement. > So, it leaves me with an ability to be presented something new, or different and very quickly I can understand it. Honestly, I think that says a lot more about your intelligence than your history. That's actually a pretty fair definition of intelligence, in fact. How about this: Teaching networking now is teaching TCP/IP and everything that TCP/IP rests upon, such as Ethernet and Wi-Fi. Theory naturally flows from practice, and that is the practice. The OSI Model? Gone. Out. Forget it. Nobody actually implements all seven layers; at most, we have four. NCP? What? If you know what NCP stands for, congratulations, you know something entirely useless. "Jeopardy!" would love you. So why teach the stuff we know is useless? It actively discourages students. It makes them think they're just wasting their time, primarily because they are. The history of computer networking is a valid topic. It should be taught in its own course, it deserves its own course. It does not deserve to be shoved into a course on how networking actually works.

glimcat 14y ago

I'm sold, although I'd still start by sitting them down with a copy of Feynman's QED. When doing math about a thing, it helps tremendously if you already have some sort of intuition about what the math is describing. Also, the historical stuff is fun and interesting. Physics students should probably be taught this stuff regardless, but maybe not as part of a class on theory.

jasomill 14y ago
On the other hand, Feynman's most important lesson is thoroughly modern: the surest way to understand something "intuitively" is to create it.
- derleth 14y ago
  > the surest way to understand something "intuitively" is to create it. And there's no reason the best way to create something has to be the way it was created the first time.
- Tichy 14y ago
  Cool idea. Screw learning QM - let's rediscover it on our own!

pirateking 14y ago

I find historical narrative to be an effective way to learn, but just reading names, dates, and important events is boring. You have to pretend like you have a time machine. An effective historical lesson takes you back in time and puts you right at the scene of the development, almost allowing you to have a conversation with the characters and follow along with their thought process. That being said, I appreciate this straightforward approach just as much, and find that having a good understanding of the core concepts first, makes adjusting the settings on the "time machine" much smoother.

derleth 14y ago
> allowing you to have a conversation with the characters and follow along with their thought process. Which were muddled, confused, and, pretty much by definition, partially incorrect. Historical development necessarily involves recapitulating the confusions of the people who finally got confused enough to throw over their original ideas. Even if you trim the history down to only the path that actually lead to modern theory, you still have a lot of confusion to get through before you can actually introduce the main idea of the topic in all its glory. And that's in addition to the confusion students have naturally.
- pirateking 14y ago
  Very true. Though there is still some value in the journey through history, aside from learning the main idea. It can serve as inspiration for your own quest, and also challenge you to question modern theory as those who came before did. Being equipped with some fundamentals first, makes exploring the history less confusing. I think Charles Petzold's book "Code", is a great example of blending historical narrative with core concepts - I highly recommend it.
Retric 14y ago
IMO, the simple approach is dangerous. Saying here is the theory have at it is fine if you want to teach someone how to design transistors, but the point is not that QM is correct it's just the best approximation yet and in no way the 'holy word' on how the universe operates. The single most important thing to teach science students is to discard existing theory's when reality does not support them and to require any new theory to also be supported by the body of existing experimental evidence.

karpathy 14y ago

I took 2 courses on Quantum Mechanics during my undergrad and I'm taking the Coursera course now as well for fun. My biggest quibble with how QM is presented is that it is, paradoxically, never tied into physics. Quantum Physics is really a Linear Algebra + Statistics but with extension to complex numbers. In my experience it is just another math course and there is no Physics anywhere. There's always talk of measuring states, applying Hadamard gates and writing down their decomposition in the eigenbasis but it's abstract and meaningless. What does a particle with some particular wave function look like? As it evolves in time it "smears" out according to the time evolution equations, but how fast is the process? Does it occur on scale of nanoseconds? Seconds? Hours? How is it suspended, or operated on? What exactly is an example of a measurement? There are so many tangible questions that would help with intuition, but are never addressed. This would be my approach to teaching quantum mechanics. Connect it to a concrete physical system, explore in detail going back and forth between experiment and math. And best, maybe even simulate the system somehow. Some time ago I made an effort to try to simulate what I learned and this was the result (as an example): http://www.youtube.com/watch?v=a88GlrUmI9Y&feature=plcp http://www.youtube.com/watch?v=a88GlrUmI9Y&feature=plcp I'ts ugly and it's probably wrong, but it's tangible and the best I could do because finding this kind of Quantum Mechanics, as opposed to a lot of talk about measuring things is very hard.

andrewcooke 14y ago
yeah, but this is more general than qm - it's amusing to ask (other) physicists how fast electrons move in typical wires (a purely classical number, but most won't have a clue).
- jules 14y ago
  It's not exactly clear what you mean by "velocity" in that question. Drift velocity? Fermi velocity? (and others) These two differ by multiple orders of magnitude, so it's easy to wrongly conclude that another physicist doesn't have a clue. Intuitive explanation: drift velocity is the overall streaming velocity of the moving electrons, fermi velocity is the velocity of a single electron. The reason these are different is that electrons move back and forth heavily, thus the fermi velocity is high. But under normal voltages, they move just slighly more in one direction than the other on average, which is the drift velocity. Of course this is highly simplified, and the real story is quantum mechanical.

gus_massa 14y ago

Perhaps this is a good approach to Quantum Mechanics for a course about Quantum Computers. (Maybe is has too much emphasis is the probability part.) Quantum Mechanics has a little set of clear rules that are easy to follow and get the correct result. So if your plan is to have some little black quantum boxes and combine them to get quantum computations perhaps it is a good idea to jump directly to the rules and forget about the history part. The problem arises when you really want to understand how the little black quantum boxes work, or how this scales to bigger systems and to the real work. The correspondence principle says that if the system is big enough the quantum effects should almost disappear, and the system should be correctly approximated by Newtonian Physic. (There are some macroscopic effects that depend on the quantum nature of the world, the most clear is superfluity.) The biggest problem is the correspondence, how to make the connection between the real world that you see and the quantum physics. The pseudo-historic path gives in each step a more complex model, but the jumps are smaller. (For example, usually the Schrödinger's model is explained before the Heisenberg's model because it is more intuitive, but Heisenberg's work was a few years before Schrödinger's work.) In this approach, there is no reference to the method that you should use to transform a classical model to a quantum model. The main examples are the harmonic oscillator and the hydrogen-like atom that are used as the base to explain the properties of the matter.

raylu 14y ago

This sounds a lot like what I started reading at LessWrong: http://lesswrong.com/lw/r5/the_quantum_physics_sequence/ http://lesswrong.com/lw/r5/the_quantum_physics_sequence/ One line that particularly stuck with me is: "Dragging a modern-day student through all this may be a historically realistic approach to the subject matter, but it also ensures the historically realistic outcome of total bewilderment. Talking to aspiring young physicists about 'wave/particle duality' is like starting chemistry students on the Four Elements."

gus_massa 14y ago
In Physics sometimes and approximated theory / model is useful, in spite it is incorrect. The problem is that the Schrödinger / Heisenberg quantum states of a particle are a lie. If you have an electron, it doesn't follow the Schrödinger / Heisenberg equation. It can emit a photon an reabsorb it a little time after. This is not part of that equation and has a very easy to measure effect that is the Lambs Shift http://en.wikipedia.org/wiki/Lamb_shift#Lamb_shift_in_the_hydrogen_spectrum http://en.wikipedia.org/wiki/Lamb_shift#Lamb_shift_in_the_hy... . And the electron can even do more crazy things, but they are luckily more difficult to measure. This is the reason to use Quantum field theory. * If you are only going to buy a lens to put in front of your photodetector, then probably the wave/particle duality is a good enough approximation (in spite that it doesn't make sense). * If you are doing quantum-chemistry then use the Schrödinger equation (in spite that it doesn't work for strong electric fields and is not compatible with special relativity). * If you work near a big particle accelerator you should try at lest to use the standard model (in spite that renormalization doesn't make sense). * And I hope that you never have to use string theory to explain and experiment.
thebooktocome 14y ago
It's nothing like the quantum physics sequence. Yudkowsky starts out making up numbers from nowhere and asserts things for purely philosophical reasons, whereas Aaronson actually has proofs (albeit left to the reader).
- loup-vaillant 14y ago
  Sure, the numbers in "Quantum Explanations" are made up, but (1) the experiments are real, and (2) everything besides the numbers is accurate (as far as I know). Plus, the goal of the sequence was never to actually explain quantum physics. It was to explain why a realist perspective (the wave function is all there is, and the math says it doesn't collapse, so it really doesn't) is by far the most probably correct, despite the fact that it makes no new prediction compared to previous interpretations. That, plus answering some philosophical question with physics. Aaronson's explanation definitely is a step in the right direction. I still have a quibble however: he keeps mentioning "probability" as an analogy to amplitude. That confuses his explanation in my opinion. I'd rather have a straight explanation of QM math, then an explanation about its similarities with probability theory. And the mixed state paragraph seems to conflate subjective probability and actual distribution of amplitude. Ick.

gwern 14y ago

> Two other perfect examples of "obvious-in-retrospect" theories are evolution and special relativity. Admittedly, I don't know if the ancient Greeks, sitting around in their togas, could have figured out that these theories were true. But certainly -- certainly! -- they could've figured out that they were possibly true: that they're powerful principles that would've at least been on God's whiteboard when She was brainstorming the world. Actually, people studying the Pre-socratics like to point out that Anaxagoras (IIRC) theorizes something eerily close to evolution: that at the beginning, there were all sorts of random creatures, and only the ones which did well survived and created more creatures like them. I don't know why noone followed up on it (in contrast to something like heliocentrism, where we know why the Greeks abandoned it for what were excellent and unobjectionable reasons at the time).

bobwaycott 14y ago

I find the references to "God" (e.g., "...why did God choose to do it that way and not some other way?") to be both distracting and unhelpful working through the text itself. It signals to my brain (possibly incorrectly) that either (1) the author is trying to inject an unwarranted religious idea into an otherwise potentially helpful explanation in a devious way, or (2) the author has sadly mis-chosen a loaded term that doesn't add anything helpful to the explanation at all, and instead detracts from working through it because it creates the nagging question in my head of, "Is he really suggesting that God (whatever that may be in the reader's mind) chose to do this?". A better option (for just the chosen example) would be: "... why does the Universe do it that way and not some other way?".

thebooktocome 14y ago
This is exceedingly common language in the mathematical community, particularly for the generation of Scott Aaronson and the one before him.
- bobwaycott 14y ago
  Oh, okay. Thank you so much for pointing that out to me. I somewhat assumed as much, and I honestly wasn't intending to create anything remotely ad hominem in my critique. Since it's not so common in hardly anything I typically read, it just stuck out strangely (I am only just digging into understanding QM/QP). Again, sincere thanks.
- andyjohnson0 14y ago
  Are they referring to a God in the religious sense, or using it as a shorthand for "nature" or "the universe"?
  - thebooktocome 14y ago
    Some of them are no doubt serious; I seem to remember a study showing that mathematicians are more likely to be theist than atheist. In my experience, "Why did God pick X?" and the like is shorthand for, "Is there a classification/uniqueness theorem that says X is the only possible solution to the problem?", which is how it is used here.