1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    fzero

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Infinite Well with Sinusoidal Potential
Loading...