Delta potential

Example

Let $m > 0$ be the mass of a particle on the line $\R$, let $\hbar$ be the reduced Planck constant, and let $\alpha \in \R \setminus \set{0}$. The delta potential is $V(x) = -\alpha\, \delta(x)$, where $\delta$ is the Dirac delta. The time-independent Schrödinger equation $-\frac{\hbar^2}{2m}\psi'' + V\psi = E\psi$ means the following. The function $\psi$ is continuous, it satisfies $-\frac{\hbar^2}{2m}\psi'' = E\psi$ for $x \ne 0$, and its derivative jumps at $0$: $$\psi'(0^+) - \psi'(0^-) = -\frac{2 m \alpha}{\hbar^2}\, \psi(0).$$

Bound state: let $\alpha > 0$, so the potential is attractive, and let $\kappa = m\alpha/\hbar^2$. Then $\psi(x) = \sqrt{\kappa}\, e^{-\kappa \abs{x}}$ is the unique normalised bound state, with energy $E = -m\alpha^2/(2\hbar^2)$. Check: for $x \ne 0$, $\psi'' = \kappa^2 \psi$, so $E = -\hbar^2\kappa^2/(2m) = -m\alpha^2/(2\hbar^2)$. The jump is $\psi'(0^+) - \psi'(0^-) = -2\kappa\sqrt\kappa = -\frac{2m\alpha}{\hbar^2}\psi(0)$. The norm is $\int_\R \kappa e^{-2\kappa\abs{x}}\, dx = 1$.

Scattering: for an energy $E = \hbar^2 k^2/(2m) > 0$, a wave coming from the left is partly reflected and partly transmitted. The transmission coefficient is $$T = \frac{1}{1 + \dfrac{m \alpha^2}{2 \hbar^2 E}},$$ and the reflection coefficient is $1 - T$. For $\alpha < 0$ the potential is an infinitely high and infinitely thin barrier, and there is no bound state. Still $T > 0$ at every energy, an instance of quantum tunnelling.