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Infinite Well with Sinusoidal Potential

  1. Feb 4, 2013 #1
    1. The problem statement, all variables and given/known data

    Assume a potential of the form [tex]V(x)=V_{0}sin({\frac{\pi x}{L}})[/tex] with 0<x<L and [tex]V(x)=\infty[/tex] outside this range. Assume [tex]\psi = \sum a_{j} \phi_{j}(x)[/tex], where [tex]\phi_{j}(x)[/tex] are solutions for the infinite square well. Construct the ground state wavefunction using at least 10 basis functions.

    2. Relevant equations

    I obtained the solution [tex]\phi_{n} = \sqrt{\frac{2}{L}} \sin({\frac{n\pi x}{L}})[/tex] for the infinite square well with zero potential inside.

    3. The attempt at a solution

    After obtaining the solution shown above, I attempted to expand the summation. I know that I want to do a linear combination of energy eigenstates, however, I am not sure what to do about the leading coefficients. I have found by searching that [tex]a_{j} = \int_{-\infty}^{\infty} \phi_{j}^{*}(x) \psi dx[/tex], but how can I solve this if I'm using the [tex]a_{j}[/tex]'s to find [tex]\psi[/tex]?

    Any help or advice would be greatly appreciated. Thanks!
  2. jcsd
  3. Feb 4, 2013 #2


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    You need to apply the Schrodinger equation to your linear combination, since the particle in a box basis are not solutions once we turn on this sinusoidal potential. You'll probably be able to use a trig identity to simplify the potential terms with [itex]V(x) \phi_k(x)[/itex], then group like orders of [itex]\sin(n\pi x/L)[/itex]. This will give you some recursive relations among the [itex]a_j[/itex]. To obtain the ground state energy, you might have to minimize something.
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