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Homework Help: Completing the square for integration of e^( )

  1. Sep 23, 2012 #1
    1. The problem statement, all variables and given/known data
    I'm working on problem 2.22 from Griffith's Intro. to Quantum Mechanics (a free particle problem). I am stuck on the final integral from part b. Part a of the problem is normalizing:
    A*e-a*x2 which I did. Part b wants the general, time-dependent wave function.

    2. Relevant equations
    Griffith says that integrals of the form
    ∫e-(a*x2+b*x) can be solved by "completing the square." Griffith gives the example of defining y to be (a)(1/2)*(x+(b/(2a)) and using that definition to convert (a*x2+b*x) to y2 - (b2/(4*a)) I presume we are supposed to substitute in the converted expression into the integral and work from there.

    3. The attempt at a solution
    I have the following integral:
    I presume using the "completing the square" method is supposed to make this integral work out; but I don't see how this will help. If I do convert this into something like:
    ey + (something about a, h, and m),
    how do I integrate with y?

  2. jcsd
  3. Sep 23, 2012 #2


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    Science Advisor

    The exponential, [itex]e^{-k^2(y^2- b^2/4a})[/itex] can be separated as [itex]e^{-k^2y^2}e^{k^2b^2/4}[/itex] and then the constant, [itex]e^{k^2b^2/4a}[/itex] can be taken out of the integral, leaving [itex]\int e^{-k^2y^2}dy[/itex]. That integral cannot be done in terms of 'elementary' functions (polynomials, fractions, trig functions, exponential, logarithms). That integral is defined to be the "Error function" or "erf(x)".
  4. Sep 23, 2012 #3
    Thank-you so much. The missing part was the change of variables.
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