Write down the general normalizable solution

[shannon]

Homework Statement

potential V(x)=-αδ(x+a)-αδ(x-a)
write down the general normalizable solution to the time independent Schrödinger equation in the case E<0, without yet imposing any boundary conditions.

Homework Equations

The Attempt at a Solution

I divided this problem into three regions x<=-a, x>=a and -a<x<a, then I got general solutions for each region, then added all those gen. solutions and got:
Ψ(x)=Aexp(Kx)+Bexp(-Kx)+[Cexp(Kx)+Dexp(-Kx)]
K=(-2mE/h)^1/2

I don't think this is right though...
can you help me out?

 

[siddharth]

Why are you adding each solution? Remember that each solution for the TISE is applicable only in those regions.

The boundary conditions would be that the wavefunction is continious and the change in the derivative of the wavefunction across the delta potential is some value.

 

[CompuChip]

You can write your solution as
$$\Psi(x) = \begin{cases} \Psi_L(x) & x \le -a \\ \Psi_C(x) & -a < x < a \\ \Psi_R(x) & x \ge a \end{cases}$$
(where $\Psi_{L,C,R}$ are solutions of the Schrödinger equation on the given domains) and that will do.