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Finding potential for a given wave function

  1. Sep 18, 2008 #1
    Here's the sitch:

    I am given an equation, A*e-a(mx2/h-bar+it)

    I need to find the value for A that will satisfy normalization, as well as find the Potential of the Schrödinger Equation using this value.

    What do I do?

    P.S. I have NOT learned gaussian integration, which is where I run into my first major problem, and this stops me from completing the problem.

    Thanks guys,
    Desh627
     
  2. jcsd
  3. Sep 18, 2008 #2
    I should also mention what I have solved so far...

    psi (x,t) = A*e-a(mx2/h-bar+it)

    psi (x,0) = A*e-a(mx2/h-bar+i(0))
    psi (x,0) = A*e-a(mx2/h-bar)

    Integral from negative infinity to infinity dx (absolute value)A*e-a(mx2/h-bar)(/absolute value)2 = 1

    A2 Integral from negative infinity to infinity dx e-a(mx2/h-bar)=1

    A = sq. rt (1/Integral from negative infinity to infinity dx e-a(mx2/h-bar))

    and this is where I run out of room.

    I should also mention that I have taken high school physics and ap calc, but no linear algebra or anything beyond ap calc.
     
  4. Sep 19, 2008 #3
    Hi,
    Let's see if we can't figure this one out.
    To figure out how to normalize your wavefunction, you need to know what the integral of a Gaussian function is, where a Gaussian function is any function of the form:
    [tex]f\left(x\right) = \exp\left(-\frac{x^2}{2\sigma^2}\right)[/tex]
    I doubt you would know a good technique from AP calculus to solve this integral, so I would suggest reading this wikipedia article:
    http://en.wikipedia.org/wiki/Gaussian_integral

    For the second part, since your wavefunction already has time explicity in it, we can make a guess that it's a solution to the time dependent schrodinger equation:

    [tex]\hat{H}\psi = i\hbar\frac{\partial}{\partial t}\psi[/tex]

    where H is the operator such that:
    [tex] \hat{H}\psi\left(x\right) = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2}\psi\left(x\right) + V\left(x\right)\psi\left(x\right)[/tex]
    And V(x) is the potential. Try applying the time-dependent schrodinger equation and see if you can convert the problem into a differential equation in x only (and not t). From there it might be possible to figure out what V(x) is.
     
    Last edited: Sep 19, 2008
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