# Double delta potential - boundary conditions

1. Mar 13, 2012

### SoggyBottoms

A double delta potential is given by $V(x) = c_+ \delta (x + \frac{L}{2}) + c_- \delta (x - \frac{L}{2})$.
Use the discontinuity relation to find the boundary conditions in $x = \pm \frac{L}{2}$.

The general solutions are:

$\psi(x) = \begin{cases} Ae^{ikx} + Be^{-ikx} & x < -\frac{L}{2} \\ Ce^{ikx} + De^{-ikx} & -\frac{L}{2} < x < \frac{L}{2} \\ Ae^{-kx} & x > \frac{L}{2}\end{cases}$

So using the discontinuity relation, we get in $\frac{L}{2}$:

$ikAe^{ik\frac{L}{2}} - ik Be^{-ik\frac{L}{2}} - ikCe^{ik\frac{L}{2}} + ikDe^{-ik\frac{L}{2}} = \Delta \frac{d\psi}{dx} = \frac{2mc_+}{\hbar^2}\psi(\frac{L}{2})$.

And that would be the first boundary condition...is any of this correct?