So let's say I have an infinite well with walls up at $x=-L$ and $x=L$. Suppose that inside the well, there is a time-dependent potential
$$ V(x,t)= \alpha_0\delta{(x)}f(t) $$
where $f(t)$ is a monotonically decreasing function. Then the time-dependent Schrodinger equation in the region $x \, \in \,[-L,L]$ is then
$$-\frac{\hbar^2}{2m}\partial_{x}^2\Psi(x,t)+\alpha_0\delta{(x)}f(t)\Psi(x,t) = i\hbar \partial_t \Psi(x,t). $$
Am I right in assuming that these are the boundary conditions? Is there anything I'm missing?
- $\Psi(-L,t)=0$ and $\Psi(L,t)=0$
- At any $t=\textrm{constant}$, $$\Delta(\partial_x \Psi(x,t)) = \partial_x \Psi(x,t) \vert_{+q} - \partial_x \Psi(x,t) \vert_{-q} = \frac{-2m\alpha_0}{\hbar^2}\Psi(x=0,t).$$
- As $t\rightarrow \infty$, $\Psi(x,t) \rightarrow \Psi_{ISQ}(x,t)$. Where ISQ stands for the infinite square well