Can a layman learn GR and QM? REALLY learn?

Factual Questions
41

Eh, I’ve seen studies of first year math grad-school test performance vs SAT scores. The basic gist is there’s a minimal (and relatively low) level of aptitude you need to be successful (I believe the term is “cognitive threshold”), but beyond that the correlation gets very weak.

So a dullard is probably not going to do well teaching themselves math, but then most dullards I know probably wouldn’t try. And if your of average intelligence or better, your enthusiasm and willingness to work at it will probably be at least as important then any innate aptitude.

42

Yes, but his only interest in physics was as a tool for picking up women.

43

There is often a misapprehension about how “smart people” learn, probably fed by montages scenes in Val Kilmer and Matt Damon films. While it is true that someone with a particular talent for mathematics mathematics may intuit concepts into application quicker, the fact is that smart people–even complete geniuses–learn the same way that everyone else does; they focus in on a topic, study it extensively, ponder and reflect upon it, test it, tease it, twist it, and finally formulate some kind of consistent set of internal rules that allow them to fit it into a larger context. The difference between “average” people like you and me, and real genuises is that they are wonks–they think about their particular field of study obsessively–and as a result are able to concoct and test novel concepts that are beyond the conventional envelope. It doesn’t hurt to be smart, but it is far more important to be devoted and fascinated by the subject.

For instance, someone with a really good mechanical intuition and a good grounding in differential equations can formulate an explicit description of a complex block-and-tackle system. But once you have that formulation, you can sit down with a combination of very simple algebraic equations of one variable and graphical aids and describe the motion of the system to an eight year old in terms they can understand. While they may not be able to go out and formulate the equations for a novel system, they’ll know enough to appreciate the mechanics and comprehend the jargon, and conceptually know that why having twice the length of rope means going half the speed.

For the o.p.: if your grounding in physics is weak or long-dormant, I would start with either Landau’s 10 Volume Course In Theoretical Physics or the venerable The Feynman Lectures on Physics. Both present physics in a way contrary to general physics textbooks, i.e. rather than going in a very linear, modular fashion through different aspects of physics, they present physics as a continuum of ideas and associated tools and methods for understanding. Both series are somewhat more conversational in tone than the standard textbook, and apply some humor. (The Feynman Lectures are actually transcribed and edited lectures that Feynman gave at Caltech as an experiment, and as such contain a lot of his characteristic humor.)

I also think highly of David Bohm’s Quantum Theory and Linus Pauling’s Introduction to Quantum Mechanics with Applications to Chemistry, both available in inexpensive Dover paperback reprints. While both are old and don’t have the latest advances in the field, they both cover the fundamentals very well, and the latter is particularly good at providing some context for the utility of QM (unlike most texts that present it as a theory standing on its lonesome). I’m currently slogging through Tony Zee’s Quantum Field Theory In A Nutshell, which is basically a survey course on QFTs. That is probably beyond a “layman” to study and learn, but is presented in a fashion that is fairly accessibly if you have a foundation in QM and wave theory and understand the notation.

Stranger

44

Did that work for him? 'Cause it ain’t doing jack for me.

“Hey, baby! How about we synchronize our quantum wave functions and get entangled?” slap

Stranger

45

FWIW, I wouldn’t recomend Landau for a first or even second exposure to a topic. They’re good, but they’re certainly not very accessible to a laymen. The Feynman lectures are a lot more accessible.

(and I just finished Zee’s book, which is a really good first exposure to QFT, though obviously one shouldn’t start learning physics with QFT.)

ETA: And using abbrivations when giving advice to someone just being introduced to a topic is kinda needlessly cryptic. QFT=Quantum Field Theory.

46

No. But learning the numerical methods and the theory behind them is almost as fascinating as the physics; for some, actually moreso. Many a promising physicist has fallen into the trap of becoming enraptured with the unpolluted beauty of computational methods. NumPy/SciPy and its ornamental priestess Matplotlib are my current computational vices.

Stranger

47

In order to really learn the subject, you would have to know a lot of mathematics that I as one holding a master’s degree in the subject am not particularly comfortable with. I know only the basic ideas and the framework in which the math is used; I could learn the actual mathematics needed to understand it if I had a good reason to learn it and a good source to learn from, but it sounds like you specifically would need to obtain a good background in graduate level algebraic concepts first. edit: to be honest though, the sort of math needed to learn this was not my favorite area of study, preferring more discrete subjects like combinatorics, number theory, and finite fields.

On the other hand, you can probably learn basic concepts of the physics quite easily. The deeper you go the more you will need the deep mathematical concepts to make sense of what you’re studying, but the surface theories are quite approachable for those with a studious mind and a desire to learn. I would personally love to know more about the subject, but I just don’t quite have the desire to find good learning sources and take the time to actually learn. When I was in school I was able to study very nonchalantly as a great deal of material just came naturally, but as time went on it got harder and I had to put serious effort in; without a degree dangling in front of me it’s hard to go on, despite how much I’d like to learn.

48

I’d actually recommend Hartle’s Gravity: An Introduction to Einstein’s General Relativity for a first pass at the subject. The focus of the first two-thirds of the book is much more on “how things behave in curved spacetime” rather than “why spacetime curves like so”; in particular, you don’t really need much more of a math background than multi-variable calculus and a bit of linear algebra to understand what’s going on in that book. Schutz is a pretty good text too, but if you just want to learn about how weird curved space-time is, Hartle is where I’d start.

49

Thanks for the tips. Feynman’s lectures are probably what got me started on all this — they’re very interesting, but the farther I got into them, the more it seemed like I needed a bit more structure to fill in the gaps. And googling around led me to the MIT OCW site, which made me realize for the first time that maybe self-study could take me all the way to grad level knowledge.

I’d be interested in reading anyone else’s picks for books that they found especially good or useful in their math or physics education.

And while I certainly appreciate the beauty of math, I was wondering if the physicists here thought that too much emphasis on the theoretical side of math might slow me down. In other words, if my primary goal is a graduate-level understanding of physics rather than math, should I stick to a Stewart or Thomas-type calculus text, rather than Apostol or Courant?

50

EXPAND UPON THIS

please

51

A few thoughts…

First, I agree that the answer is “yes”, assuming sufficient motivation on the learner. However, while instructor-less education is certainly going to be a little harder, I think it is more harder than it may seem at first. There will be many things presented in the chosen texts that just don’t make sense. In other words, you will get stuck. In a formal setting, there would be instructors (or fellow students) to provide personalized clarification. This aspect of learning advanced material is more important than it might seem, because these roadblocks can come and go in an instant in a formal setting – so quickly that you don’t even notice:

  • “Wait, why does this extra term appear here? I’m getting a different result…”
  • “Because the particles are indistinguishable, remember, so you have to divide by N! earlier, leading to this extra term in the logarithm.”
  • “Oh, right, of course! :smack:”

Wham, bam, confusion averted. In an isolated setting, trivial things like this can hinder progress indefinitely or cause you to move ahead prematurely. Having more than one reference source per topic will help some, but it will still be the case that learning advanced material will be significantly less time-efficient when done alone because there is no one to bounce thoughts off of or to learn from. (There’s always the SDMB, I suppose!)

Another point: much of your time will be (or, rather, should be) spent working problems. It takes little time to read a chapter; it takes more time to work through enough relevant problems so that the tools and concepts get internalized in a way that reading alone does not induce. I’ve seen people try to learn advanced material in their spare time, and they always skimped on working problems, and they always got stymied by something soon after getting started.

52

I’ve also seen the opposite, though. My mom’s currently taking a physics course, and so over Christmas break, I gave her a crash course in trig. The second evening of it, she said “Oh, and I also put together all these homework problems for myself; did I do them right?”.
Frylock, the key to the Uncertainty Principle is that it’s really a statement about position and wavelength, not position and momentum: It’s just that in quantum mechanics, there’s a relationship between momentum and wavelength. But the uncertainty relationship between position and wavelength shows up in all waves. Take a rope, tie one end to something, and hold the other end and make waves on it. You can send a single sharp pulse down the line, and at any given moment, that pulse will have a very well-defined position, but the concept of “wavelength” doesn’t really apply to it, since it’s not periodic. The position is well-defined, but the wavelength isn’t. On the other hand, you can continually wave the rope back and forth, and make something that has a very clear wavelength, but now, you can’t really say where the wave is: It’s everywhere. Or you can do something that’s somewhere in between, but there’s still a relationship, there: The better you know the position, the worse you know the wavelength, and vice-versa. This is clearer in the math, but that’s the gist of it.

53

Funny, because I just realized that today. I was looking at an example in my Calc text, trying to do anticipate the next step, and I didn’t see where to go. So I gave up after a few minutes and looked at the next line, and it was a very simple trig substitution.

And for whatever reason, it just became so clear to me that reading through an example, even if you are sure you understand every step, is MUCH easier than doing it yourself, where you have to figure out what to do next, and not just understand what was done for you.

I guess that should have been obvious, but for some reason I never fully appreciated it until now. So I’m going to slow down and do more problems.

54

Okay, and do you promise on your honor cross your heart that Heisenberg’s Uncertainty Principle is just a straightforward and direct consequence of the very same math, once you throw in the wavelength/momentum relation?

I ask because I’m going to go around telling people this now and I want to make sure I’m right. A sufficiently solemn promise, perhaps with hand raised, should do the trick I think.

55

It’s interesting, isn’t it, that the study of the fundamental building blocks of our universe (what in other fields we may call primitives), by people who prize elegance and simplicity of theories, turns out to be arguably the biggest mindfuck yet for the human race.

But I guess there aren’t many fields that make such incredibly precise predictions or require so much extrapolation just to get data. Or where intuition is (often) such a hindrance.

</late_night_post>

56

Einstein, for one, claimed that this was evidence that our theories were not yet good enough. He felt that a complete theory of physic primitives would indeed be dimple and elegant.

57

Um, if you’re going to go around and espouse this, you should probably know enough about to explain why it is true. Jim Al-Khalilil’s Quantum: A Guide for the Perplexed is probably the best non-technical description I’ve seen of quantum mechanics, and if memory serves he spends a significant amount of the book on indeterminacy and how it relates to classical wave theory, as well as thoroughly debunking any mysticism about the observer’s effect on the two-slit experiment or any business about cats and friends in superposition.

Stranger

58

Any sufficiently advanced theory should indeed have dimples. And maybe freckles.

Stranger

59

My position is perhaps a little different from the OP’s, but I have a question which I think this thread is still relevant to: so far as the mathematics goes, I understand linear algebra, Hilbert spaces, Fourier theory, and all that. I have, I think, all the requisite mathematical background to understand basic quantum mechanics (or at least, enough that I could quickly familiarize myself with whatever I need).

And so, when I read things like how “states” correspond to lines in some Hilbert space, with “observables” corresponding to linear operators on that space, the eigenvalues of which are possible “measurement results”, etc., etc., I understand the mathematical bits just fine. I know what those words refer to. But I still don’t understand how to connect up the “states” and “observables” and so on with anything in the real world. I’m never able to figure out how the mathematics of quantum mechanics connects with an actual physical situation. I have no idea why all this mathematics is relevant to quantum mechanics (other than just knowing by fiat that it is), and how to translate back and forth between the two.

It’s as if I understood calculus and someone told me “force is mass multiplied by the second derivative of position with respect to time, etc., etc.”, but I had no ordinary language understanding of what words like “force”, “mass”, “position”, and “time” amounted to, so all of Newton’s laws were just meaningless definitional tautology to me; I could understand the math just fine, but I would have no idea how to interpret it into actual claims about the behavior of moving bodies.

As a contrast, for general relativity, which I also don’t know anything about, it seems easy enough to understand how the math and the physics matches up; spacetime is some manifold, inertial paths can be thought of as its geodesics, the phenomenon of gravity amounts to some equation describing the relationship of its curvature to mass, bam. It’s easy enough to see why and how you would use differential geometry to model all this.

But with quantum mechanics, I have basically no idea why and where Hilbert spaces and self-adjoint operators and all that come up in the first place; I just know that they do because I keep hearing that they do. I have no idea what situations involve Hilbert spaces and self-adjoint operators, what spaces and operators to use in those situations, what to do with them, how to interpret the results in empirical terms, and why physicists were led to work in this whole “Measurement results are eigenvalues, etc., etc.” formulation in the first place.

I guess I’m being both rather repetitive and rather vague about where my confusions lie, but, to take one last inarticulate stab, it’s something like this. Over at this Wikipedia link, you can find a selection of what are apparently the basic postulates of quantum mechanics. I understand all the mathematical words in them, but I have no idea what any of this has to do with the behavior of particles in the world. “Every physical system is associated with a topologically separable complex Hilbert space with inner product…” Well, that’s great, but it doesn’t tell me anything about what the actual association is. And so on for all the rest of it. It’s like a long list of “Here’s some mathematics that can be applied” without a clear explanation of quite how to apply it.

So, my question is… what’s a good resource for someone like me to patch up this deficit and learn how to interpret the mathematics in quantum mechanical terms? Whenever I go looking for a “Quantum mechanics without the math ripped out”* primer, I never make it beyond the same sort of state that Wikipedia link leaves me in. I always feel like if I just had some physicist friend who I could ask questions of in person, they could explain these things to me quite quickly, but alas, I have only this board. So, uh, point me in a helpful direction.

*: And of course, “quantum mechanics with the math ripped out” just means “quantum mechanics with the quantum mechanics ripped out”, which is not of any use to me either…

60

“Um” I figured I had the gist from Chronos’s explanation. A wave packet has no definite wavelength, and has a definite position. A standing wave has a definite wavelength, and no definite position. QT marks out an important relation between wavelenth and position such that this relation between definability of position and wavelength, in QT, implies a similar relation between definiability of position and momentum.

That’s pretty much right, right?

I’d love to understand it with more depth, but for lay conversations what I just said–assuming it’s right–looks pretty good. It certainly represents a deeper understanding of the UP than almost any non-phycisist has, so I can’t see what’s wrong with a non-physicist saying it.

I’ll read the book you mentioned with great enjoyment and I’ll learn alot from it, but not because I think what I just said was somehow clueless, meriting remarks beginning with smart-alec internet-nerdy “um”'s and everything.

Rather, I think what I just said above was clueful, yet still quite shallow, yet better than what I had before and better than what anyone I know has yet. So I figured I’d share.

ETA: I have not been under any impression that observer effects have anything to do with this stuff. I don’t know why you projected that onto me.