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Does it take two years of calculus to understand Schrodinger's equation

  1. Jan 17, 2012 #1
    I'm getting ready to teach myself calculus with the ultimate goal of then trying to learn as much advanced physics I can in about a year's time. I've heard conflicting views about how hard this stuff is. One book said it takes two years of calculus before one can understand Schrodinger's equation. Michio Kaku, on the other hand, says physics gets easier as one goes along. I have a feeling though that he was saying that because he was throwing a sales' pitch to people to get involved in science. Another friend of mine said calculus is easy. I understand the question is rather meaningless, but how hard is calculus and advanced physics? I realize its meaningless because you can't really compare it to anything but I would still like to hear some ideas.

    I put this in the quantum m category because it specifically references Schro equation.
     
  2. jcsd
  3. Jan 17, 2012 #2
    Advanced physics and mathematics is not hard at all if you have the capability of looking at things from different point of views and compare them with each other.
     
  4. Jan 17, 2012 #3

    Ken G

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    Gold Member

    I don't think the Sh. eq. requires two years of calculus to understand, but it might require that to solve it in novel situations. Indeed, for that it's probably even more important to have a deep understanding of algebra (in the way mathematicians use the term) than calculus! But even without that you can begin to understand it if you understand its structure, without knowing how to solve it in general situations. The full Sh. eq. is a partial differential equation because it depends on both x and t, but the usual trick to solving it reduces it quickly to an ordinary differential equation that depends only on x, which knocks a year off the calculus right there!

    The trick is to look for "energy eigenfunctions", which have "stationary" behavior (which really means their magnitude stays fixed and their phase advances linearly in time, like the hands of a clock). You can then build up any general solution as a linear combination of eigenfunctions that are each just going around like clocks, but they have complicated spatial behavior that can do whatever you need when you superimpose them to fit the initial conditions you are given. The spatial behavior is what you need calculus for-- you replace the full Sh. eq. with the "time independent" Sh. eq. by asserting you are just solving for the spatial dependence of each "energy eigenfunction". That means you can replace d/dt by -omega (that's the clock business) and the rest of the Sh. eq. is now an ordinary differential equation in x, for the eigenfunction that corresponds to that omega. It might not work for just any omega, but when it works only for discrete omega, algebra theorems tell you that you have a discrete spectrum of eigenstates that you can expand any general solution in terms of. So finding those eigenfunctions (functions of x) is the tricky part, but you can just look at the tricks used, you don't really need to be able to solve it yourself to understand either what the equation is doing, or what is the nature of the eigenfunction. That is already a pretty useful form of understanding of the Sh. eq.
     
  5. Jan 17, 2012 #4
    Not quite sure what you mean by 2 years, do you mean 5 courses? Well for most physics majors yeah, that's the starting point. You need about 3 classes in calculus, 1 in linear algebra, 1 in differential equations. You need to know linear algebra and be familiar with how to solve a complex (i.e. something with an imaginary component) diffusion differential equation.

    As for difficulty it's impossible to say without knowing your baseline. No one would say physics is easy, but it you're motivated it's perfectly surmountable. You have to put work into it, work through problems, practice and stay motivated.
     
  6. Jan 17, 2012 #5
    I'm currently auditing a class in QM but have an engineering background and not a physics background. I actually find the material rather simple and strait forward, however I'm getting a PhD in engineering which is really just a PhD in applied mathematics.

    In order to completely understand QM at the introductory level I would say that you need to be proficient in multivariable calc, diff-eq, linear algebra, and complex analysis. As mentioned above a lot of it is just solving differential equations. So if you can solve the beam or membrane equation or even the NS equations, you should be fine. I think this takes more than a few classes in calc though.
     
  7. Jan 18, 2012 #6
    I guess we need to know a few things:

    (1) What do you know already?
    (2) What do you want to know/gain from learning all this maths and Physics?
    (3) How much time are you willing to invest in learning what it is you think you want to learn?
     
  8. Jan 18, 2012 #7

    jtbell

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    Did you ever take an "introductory modern physics" course like this one?

    https://www.amazon.com/Modern-Physi...=sr_1_1?s=books&ie=UTF8&qid=1326871761&sr=1-1

    Or was your first exposure to QM via a full-on QM textbook like Griffiths?

    I taught a second-year intro modern course for many years, using that book and one by Beiser. The only mathematical prerequisites were basic differential and integral calculus, including calculus of trig functions, chain rule, basic substitutions for solving integrals, etc. Physics prerequisites were classical physics via the usual two-semester calculus-based intro physics course.

    It taught/reviewed enough basic stuff about complex numbers and partial derivatives to get started, and introduced students to the basic idea of a differential equation.

    The only examples that we "solved" in detail were piecewise-constant potential ones like the particle in a box, the finite square well, and the rectangular barrier, using the hand-waving "guess and try" method: "OK, we want a function whose second derivative is a constant times the function itself, what are the candidates?" Most of the work in those problems is in meeting the boundary conditions.

    The student has to accept a bit of "hand-waving", but it's a start.
     
    Last edited: Jan 18, 2012
  9. Jan 18, 2012 #8
    Thanks for the info I really appreciate it. Some people asked for some personal info, like what my degree of motivation is, why do I want to learn it, and how much do I know already. I think I'm going to start a blog to answer those questions.
     
  10. Jan 18, 2012 #9

    Vanadium 50

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    Solving the hydrogen atom surely takes that much. At least. If you want to solve it in parabolic coordinates, that takes even more mathematical sophistication.
     
  11. Jan 18, 2012 #10

    jtbell

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    Staff: Mentor

    Sure, actually deriving the whole solution for the hydrogen atom from scratch takes a lot of work. I didn't see all the pieces (including deriving the associated Laguerre polynomials) until graduate school.

    The original question was about "understanding Schrödinger's equation." The starting point for that is the typical one-dimensional problems that you do in a second-year modern physics course. Even in my second-year course we at least did separation of variables in spherical coordinates for the hydrogen atom. At that point we simply presented the spherical harmonic functions and some of the radial functions, and verified that they are indeed solutions.
     
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