I have a question about the problem with a particle in a box and an additional delta potential:
$V(x)=g\delta(x)+V_B(x)$
with $V_B(x)=0$ for $|x|\leq L$ and $V_B(x)=\infty$ elsewhere.
From this I want to determine the wave functions of the ground state and the first excited state in the whole space with $g<\infty$.
Now, my ideas. The normal approach for a particle in a box is:
$\Psi(x)=A\cos(kx)+B\sin(kx)$,
or, equivalently, $\Psi(x)=Ae^{ikx}+Be^{-ikx}$,
with the boundary conditions
$\Psi(\pm L)=0$.
This approach is used in the Schrödinger equation to determine the energy eigenvalues.
Furthermore, I know this condition for delta potentials (from integrating the Schrödinger equation):
$\Psi'(0^+)-\Psi'(0^-)=\frac{2m}{\hbar^2}g\Psi(0)$
My problem is, how does the delta potential change the problem, so how do you proceed in this case?
Please note: I do not require a complete solution for some homework, or anything of that kind. I would just like to know, how the delta potential changes matters, and which Ansatz for the wave function is used here.
