Why is the wavefunction not equal to 0 at the spike of the Dirac delta potential, but it is 0 for the infinite square well boundaries?

Question

Let an infinite square well have its left corner at origin and right corner at $x=L$. We then have the following boundary conditions for a wavefunction $\Psi(0)=\Psi(L)=0$. Now for a different system let $V$ be the potential $V(x)=\lambda \delta (x)$ where $\delta(x)$ is the dirac delta function and $\lambda >0$. The general solution (according to my physics professor) is for a particle coming from the left

$$\Psi(x)=\begin{cases} Ae^{ikx}+Be^{−ikx} &\text{ for }x \le 0 \\ Ce^{ikx} &\text{ for }x \ge 0 \end{cases}$$

and he states "the physical solution must be continous at $x=0$. I think I agree with that however when I asked him "Why don't we have the boundary condition in this case $\Psi(0)=0$ due to $\delta(0)=\infty$ just as in the case with the infinite square well. He answered something but I didn't really understand. And could the solution be as following?

$$\Psi(x)=\begin{cases} Ae^{ikx}+Be^{−ikx} &\text{ for }x < 0 \\ Ce^{ikx} &\text{ for }x > 0 \\ 0 &\text{ for }x = 0 \end{cases}$$

So my question still remains. Thanks in advance.

Answer 1

Keep in mind that the Dirac delta function is not actually a function. It is typically introduced to physics students as an "infinite spike" such that for if $f(x)=\delta(x)$ then $f(0)=\infty$. This is not very rigorous, and it leads to confusions like the one you are facing now.

Instead, consider the Dirac delta "function" as the limit of a function that has a spike of finite height and width at $x=0$ as we cause the spike to get taller and thinner while still requiring the area under the spike to be constant. Then you will see that the spike is just part of the potential; it is not a boundary condition.

Remember, the reason we don't want a non-zero wavefunction in the infinite potential region of the square well is because it is a non-infinitesimal region with infinite potential. This is not the case for the above limit, as the potential will only approach infinity at a single point, so we have no issues with not setting $\psi(0)=0$ where the Dirac delta function is centered.

As for your proposed solution, if you were to work out the rest of the problem you would find $A=B=C=0$, so the only way $\psi(0)=0$ is if you have no particle at all.

Answer 2

Physical wave functions must have finite energy, but it is not always the case that $V(x) = \infty \Rightarrow \psi(x) =0$

For the infinit square potential, if $\psi$ is non zero outside the well, then $\int \psi^* H\psi$ will be infinite.

For the $\delta$ potential (or the Coulomb potential, for that matter), there are wave functions $\psi$ with $\psi(0) \neq 0$ and $\int \psi^* H \psi < \infty$.

This is because $\int {\rm{d}} x \psi^* (x) \delta(x) \psi(x) = |\psi(0)|^2$.