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Solving for Cn to get wave function

  1. Dec 7, 2016 #1
    I need to solve Cn for a wave function, and have reached the following integral:

    Cn = -[√(1/a)](a/nπ)[cos(nπx/a)(ψ1(x)+ψ2(x))+∫cos(u)(dψ1(x)/dx)dx+∫cos(u)(dψ2(x)/dx)dx]


    This is a simplified version of the original equation, for
    elaboration Cn is the constant for linear combinations of a wave function. This is also strictly for a infinite stairwell wave function. I need help solving this.
     
  2. jcsd
  3. Dec 7, 2016 #2

    Simon Bridge

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    Lets see if I can tidy that up:
    ##c_n = -\left[\frac{1}{\sqrt{a}}\frac{a}{n\pi} (\psi_1(x)+\psi_2(x))\cos n\pi x/a + \int \cos u \frac{d}{dx}\psi_1(x)\; dx + \int\cos u \frac{d}{dx}\psi_2(x) \; dx \right]##
    ... that cannot be correct (unless ##c_n## is a function of x): have you missed out an integral sign there?
    ... when you change variable, you need to express the whole integral in terms of the same variable.
    ... you cannot solve the integrals given without an explicit form for ##\psi_1## and ##\psi_2## ... or some indication of what they are supposed to be;
    ... when presenting a problem, please show us the original problem statement. It looks like you already have a linear combination of wavefunctions, so it is unclear what this ##c_n## is supposed to do. I have a feeling you are going about your task the wrong way.

    I mean; I can see terms that look like ##\phi_n = A\cos n\pi x/a##
    ... which, for n=1,3,5... and ##A=\sqrt{\frac{2}{a}}##, are energy eigenfunctions for an infinite square well, width ##a##, centered on the origin. Maybe you used ##\phi = A\sin n\pi x/a## and what I see is the result of an attempt at integration by parts?

    You often need to integrate by parts twice to get something useful.
    Note: ##\frac{d^2}{dx^2}\psi_n = -\frac{2m}{\hbar^2}\big(E_n - V(x)\big)\psi_n## because Schrodinger.

    Note: to do QM, you really really need LaTeX.
     
    Last edited: Dec 7, 2016
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