Eigenvalues and Eigenfunctions for a Delta Potential Well

[Timelord_Zhou]

Homework Statement

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.

Homework Equations

i can only reder to stationary Schrodinger equation.

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.

 

[TSny]

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.

 

[Timelord_Zhou]

thanks