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Delta potential well question

  1. Nov 16, 2012 #1
    1. The problem statement, all variables and given/known data

    Consider this situation, V(x)=λδ(x) ,-a<x<a. V(x)=∞,x>a or x<-a.
    How to find the eigenvalue and eigen wavefuntion of the Hamiltonian.

    2. Relevant equations
    i can only reder to stationary Schrodinger equation.

    3. The attempt at a solution
    when it is ouside the well(x>a or x<-a), the wave function is zero.
    But when it is inside the well, i just do not know how to solve the Schrodinger function because i do not know how to deal with the dela potential.
     
  2. jcsd
  3. Nov 16, 2012 #2

    TSny

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    Gold Member

    Welcome to PF!

    The Schrodinger equation may be written as

    ##\psi '' (x) = -\frac{2mE}{\hbar^2}\psi(x) +\frac{2m\lambda }{\hbar^2}\delta (x)\psi(x)##

    The effect of the delta function is to impose a boundary condition on the slope of the wavefunction at ##x = 0##; i.e., a boundary condition on ##\psi'(x)## at ##x = 0##. You can discover the specific form of this condition by integrating both sides of the Schrodinger equation from ##x = -\epsilon## to ##x = \epsilon## and then letting ##\epsilon## approach zero.
     
  4. Nov 20, 2012 #3
    thanks
     
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