Eigenvalues and Eigenfunctions for a Delta Potential Well

AI Thread Summary
The discussion focuses on finding the eigenvalues and eigenfunctions for a delta potential well defined by V(x)=λδ(x) within the range -a<x<a and infinite outside this range. The stationary Schrödinger equation is referenced, highlighting that the wave function is zero outside the well. Inside the well, the challenge lies in handling the delta potential, which imposes a specific boundary condition on the wavefunction's slope at x=0. This boundary condition can be derived by integrating the Schrödinger equation around the delta function and taking the limit as the integration bounds approach zero. Understanding this condition is crucial for solving the problem effectively.
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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.
 
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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.
 
thanks
 
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