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Homework Help: Schrodinger Equation help

  1. Dec 7, 2007 #1
    1. The problem statement, all variables and given/known data
    The wave function of a particle satisfies the time-independent schrodinger equation.
    If the potential is symmetric and has the form
    [tex] V(x) = \inf[/tex] |x|>1.0
    [tex] V(x) = \frac{\hbar^2V_0}{2m} [/tex] |x|<0.2
    [tex] V(x) = 0 [/tex] Elsewhere
    Using the shooting method, I need to find the ground state energy and the normalised group state wave function if [itex] V_0 = 50 [/itex]. What is the energy of the first excited state?


    2. Relevant equations



    3. The attempt at a solution
    I have no idea where to even start. I only have the very basic of ideas of how the shooting method works. I am suppose to program this, but the coding shouldn't be a problem. I just don't even know where to begin. I do know from that the boundary conditions are [itex] \phi(-1) = \phi(1) = 0 [/itex]. Other than that I'm clueless.

    edit: [itex] \phi [/itex] represents the wavefunction, I just dont know how to write it.
     
  2. jcsd
  3. Jan 17, 2008 #2
    Hi I'm a new member. Where can I find out numerical method for solving Schrodinger equation? Thanks
     
  4. Jan 17, 2008 #3

    Dick

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    phi(1)=0 is correct. phi(-1)=0 is not. phi(-infinity) needs to be zero in the limit to make the wave function normalizable. The time independent Schrodinger equation is an second order ode for phi(x). So your initial values at x=1 are phi(1)=0 and phi'(x)=c. You pick c and integrate backwards towards -infinity. Find values of c so that phi blows up to +infinity and then to -infinity. Keep splitting the difference until you find one that's relatively stable. You won't find an exact one. You can only guess an estimate. The methods of evolving a second order ode are pretty standard. I usually lean towards a simple predictor-corrector method. Google it.
     
  5. Jan 18, 2008 #4

    malawi_glenn

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    It depends on what the potential are etc. Shrodinger is a differential eq, so search for numerical methods for solving those.

    I have never solved the SE nummerical with anything else than matlab's diff eq solver. Which is a Runge-Kutta method.
     
  6. Jan 18, 2008 #5

    mda

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    Are you sure? it is a symmetric potential with infinite boundaries
     
  7. Jan 18, 2008 #6

    malawi_glenn

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    according to me, your original posted boundry conditions are correct.

    why cant you solve this analytically? It is not a hard diff-eq to solve.
     
  8. Jan 18, 2008 #7

    Dick

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    Ooops, you guys are right. Somehow I missed the absolute values in V(x)=inf for |x|>1. But I'm still not sure you can do it analytically, can you? You can still use shooting to approximate the answer by trying to hit phi(-1)=0 numerically.
     
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