Quantum Tunnelling with Delta Potential

Question

I'm trying to create an animation of Quantum Tunnelling like this one.

I've been learning some QM on my own, so please forgive and correct any mistakes.

I considered the potential barrier $\alpha \delta(x)$ where $\alpha $ is a real constant and $\delta$ is Dirac's.

I assumed a wave coming in from the left (travelling to the right), that either reflects off the barrier or tunnels through the barrier. Solving the time independent Schrodinger equation gave me $$\psi(x) = \left\{ \begin{array}{ccc} 1\mathrm e^{\mathrm ikx} + R\mathrm e^{-\mathrm ikx} & : & x<0 \\ T\mathrm e^{\mathrm i kx} & : & x > 0 \end{array}\right.$$ where $k = \frac{\sqrt{2mE}}{\hbar}$. Here $|R|^2$ gives the probability of the wave reflecting and $|T|^2$ the probability of the wave tunnelling through the barrier.

We want $\psi$ to be continuous and we want, as $\varepsilon \to 0^+$, $$-\frac{\hbar^2}{2m}\int_{-\varepsilon}^{\varepsilon}\frac{\mathrm d^2\psi}{\mathrm dx^2}~\mathrm dx+\alpha\int_{-\varepsilon}^{\varepsilon}\delta(x)\psi(x)~\mathrm dx= E\int_{-\varepsilon}^{\varepsilon}\psi(x)~\mathrm dx$$ $$\lim_{\varepsilon \to 0}\left[\frac{\mathrm d\psi}{\mathrm dx}\right]_{-\varepsilon}^{\varepsilon} = \frac{2m\alpha}{\hbar^2}\psi(0)$$ Applying these conditions gives me $$R=\frac{\alpha}{2\mathrm ik-\alpha} \ \ \ \mbox{ and } \ \ \ T=\frac{2\mathrm ik}{2\mathrm ik-\alpha}$$

$$\psi(x) = \left\{ \begin{array}{ccc} \mathrm e^{\mathrm ikx} + \left(\frac{\alpha}{2\mathrm ik-\alpha}\right)\mathrm e^{-\mathrm ikx} & : & x<0 \\ \left(\frac{2\mathrm ik}{2\mathrm ik-\alpha}\right)\mathrm e^{\mathrm i kx} & : & x > 0 \end{array}\right.$$

Including the time-dependent term $\varphi(t)=\mathrm e^{-\mathrm iEt/\hbar} = \mathrm e^{-\mathrm ik^2\hbar t/2m}$ gives

$$\psi(x)\varphi(t) = \left\{ \begin{array}{ccc} \mathrm e^{\mathrm ik(x+k\hbar t/2m)} + \left(\frac{\alpha}{2\mathrm ik-\alpha}\right)\mathrm e^{-\mathrm ik(x-k\hbar t/2m)} & : & x<0 \\ \left(\frac{2\mathrm ik}{2\mathrm ik-\alpha}\right)\mathrm e^{\mathrm ik(x+k\hbar t/2m)} & : & x > 0 \end{array}\right.$$

I've looked at $|\psi(x)\varphi(t)|^2$ and this is independent of $t$.

Griffiths mentions taking a linear combination of the $\psi(x)\varphi(t)$, but does not give any details.

Any ideas?

Answer 1

If what you wanted was to get something that changes in time, then you started off on the wrong foot when you went looking for solutions for the time-independent Schrödinger equation. The wavefunction you have written down is an eigenfunction of the hamiltonian, and as such, no physical observable will ever change in time.

If what you want is to construct a solution with a wavepacket that actually moves, then that's never going to be a solution of the TISE; instead, you need to build a solution of the time-dependent Schrödinger equation, with a suitable initial condition, and then let that propagate.

Luckily, you've already done most of the required work, in building out the relevant continuum eigenstates $\psi_k(x)$ (and therefore their associated TDSE solutions, $e^{-i\hbar k^2 t/2m}\psi_k(x)$), and all you need is to assemble those into a wavepacket. The way that's normally done is by starting with a gaussian on the left and with momentum to the right, $$ \psi_0(x,t_0) = N \exp\left(-\frac{1}{2\sigma^2}(x-x_0)^2+ip_0x\right), $$ decompose that via a Fourier transform into a sum of plane waves, extend those plane waves into the barrier eigenstates you've found, add the time-dependent phase, and then do the Fourier transform back into position space.