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Mathmetics in Introductory Quantum Mechanics book

  1. Oct 5, 2011 #1
    I'm reading Introductory quantum mechanics written by liboff.
    When I solve problems, I stuck with calculation such as Intergral(-inf to +inf) e^(-x^2)dx, Intergral(-inf to +inf) (x^2) e^(-x^2)dx, and other many integrals.
    I studied thomas' calculus but I think I haven't seen these in the book. So can't do it. Are there math books about these integrals? What math subject is related to these (high level?)integrals?
     
    Last edited: Oct 5, 2011
  2. jcsd
  3. Oct 5, 2011 #2

    jtbell

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    These are examples of Gaussian integrals:

    http://en.wikipedia.org/wiki/Gaussian_integral

    I don't recall seeing them in any of my undergraduate math courses. I think most physics students (in the USA at least) learn about them in their physics courses.
     
    Last edited: Oct 5, 2011
  4. Oct 5, 2011 #3

    dextercioby

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    <Intergral(-inf to +inf) e^(x^2)dx> and the other one. I think you meant them with a minus e^(-x^2).

    It depends on the curricula. An advanced course of calculus is normally taken before quantum mechanics or statistical mechanics.
     
  5. Oct 5, 2011 #4

    HallsofIvy

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    The first one, Integral(-inf to +inf) e^(x^2)dx, is not in Thomas because it does not converge. But [itex]\int_{-infty}^\infty e^{-x^2}dx[/itex], as dextercioby suggests, certainly is in Thomas, and the second is a variation. You may be trying to find an anti-derivative formula and not finding that- neither integrand has an "elementary" anti-derivative. Both are used extensively in probability ([itex]e^{-x^2}[/itex] is the "bell shaped curve") and so in quantum mechanics.

    Here is a simple way to get the first integral:
    Let [itex]I= \int_{-\infty}^\infty e^{-x^2}dx[/itex]. Since the integrand is symmetric about x=0, we also have [itex]I/2= \int_0^\infty e^{-x^2}dx[/itex].

    And, we can write [itex]I/2= \int_0^\infty e^{-y^2}dy[/itex]. Multiplying those together, [itex]I^2/4= \left(\int_0^\infty e^{-x^2}dx\right)\left(\int_0^\infty e^{-y^2}dy\right)[/itex]. By Fubini's theorem, we can write that product of integrals as a double integral:
    [tex]I^2/4= \int_{x= 0}^{\infty}\int_{y=0}^\infty e^{-(x^2+ y^2}dydx[/tex]

    Now, change to polar coordinates: [itex]x^2+ y^2= r^2cos^2(\theta)+ r^2sin^2(\theta)= r^2 and [itex]dydx= r drd\theta[/itex]. The area of integration, with both x going from 0 to infinity is the first quadrant. In polar coordinates, r goes from 0 to infinity while [itex]\theta[/itex] goes from 0 to [itex]\pi/2[/itex]. The integral becomes
    [tex]I^2/4= \int_{\theta= 0}^{\pi/2}\int_{r=0}^\infty e^{-r^2} rdrd\theta= \frac{\pi}{2}\int_0^\infty e^{-r^2} rdr[/tex]

    That extra 'r' in the integrand now allows us to make the change of variable [itex]u= r^2[/itex] so [itex]du= 2r dr[/itex] and the integral becomes
    [tex]I^2/4= \frac{\pi}{4}\int_0^\infty e^{-u}du[/tex]
    which is easy.

    It was necessary, to make that change to polar coordinates, that the integral be over the entire first quadrant. Again, that function, [itex]e^{-x^2}[/itex], has no elementary anti-derivative. In fact, its anti-derivative is typically written "erf(x)", the "error function", and it values are got by a numerical integration.
     
    Last edited: Oct 5, 2011
  6. Oct 5, 2011 #5
    Thank you very much for your helps and I'm glad to finally locate it.
    Trying to find that integral, I have flipped over pages of thomas so many times.
    Even I borrowed from library advanced calculus written by bucks. I found gamma function and Integral(-inf to +inf) e^(-x^2)dx in this book. Now i'm about to read both of them. Thanks again.
    P.S. I'm curious about how to input mathematical notations at the post.
     
  7. Oct 5, 2011 #6

    jtbell

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    https://www.physicsforums.com/showthread.php?t=386951 [Broken]
     
    Last edited by a moderator: May 5, 2017
  8. Oct 5, 2011 #7

    dextercioby

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    For the records, just because it's <Introductory> QM, it doesn't mean it uses simple mathematics. That's why serious professors always provide their students with the so-called <prerequisites> before attempting any of their courses. As anybody should know, complex analysis and multi-variable differential and integral calculus are pre-requisites for a quantum mechanics course.

    Of course, books don't have prerequisites, but merely opening one should get you informed on the necessary mathematics you need to comprehend its content.
     
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