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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?

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  • $\begingroup$ Solving the time-independent Schrodinger equation gives you eigenfunctions, which by definition don't change with time (except up to a phase). The correct thing to do is either start with an initial wavepacket and solve the time-dependent Schrodinger equation numerically, or Fourier expand the wavepacket in terms of your eigenfunctions. $\endgroup$ – Jahan Claes Jul 27 '17 at 21:02
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    $\begingroup$ As an aside, using the notation $\theta(t)$ in this context for anything other than the Heaviside step function can be a bit confusing - it's best avoided unless there's a good reason. $\endgroup$ – Emilio Pisanty Jul 27 '17 at 21:26
  • $\begingroup$ @EmilioPisanty Thank you - I've changed $\theta$ to $\varphi$. $\endgroup$ – Fly by Night Jul 27 '17 at 22:00
  • $\begingroup$ @JahanClaes I'm not really sure what you mean by a wave packet. Is it what we get by summing $\psi_k(x)\varphi_k(t)$ for $k$ in an open interval? $$\int_{k \in I} f(k)\psi_k(x)\varphi_k(t)~\mathrm dk$$ You mention the Fourier expansion. I don't know if this is related, but if I have $\Psi(x,0)$ and I want to find $\Psi(x,t)$ then I use the Fourier Transform like Griffiths on pages 61 and 62. $\endgroup$ – Fly by Night Jul 27 '17 at 22:04
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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.

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  • $\begingroup$ As an aside; odds are you're going to be hard pressed to find a nice tractable elementary solution to your time dependent wave-function at the end of all of this. I've tried something like this before, and the integrals got quite messy and difficult to deal with. It was honestly quicker for me to use maple to numerically solve the time dependent Schrodinger equation, approximating the delta potential with a sharply peaked Gaussian. $\endgroup$ – chuckstables Jul 27 '17 at 21:44
  • $\begingroup$ Thanks for your reply. As you say, my eigenfunctions are not physical states - they're not normalisable. Griffiths (on page 75) makes that point, and goes on to say that we need to make a linear combinations of these states. I thought that maybe it would be an integral over a short range of $k$ values, say $$\int_{K-\varepsilon}^{K+\varepsilon} f(k)\psi_k(x)\varphi_k(t)\mathrm e^{\mathrm ixk}~\mathrm dk$$ I am probably totally wrong though! $\endgroup$ – Fly by Night Jul 27 '17 at 22:11
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    $\begingroup$ If I had to guess I'd say it was a numerical simulation. I have a java program that I downloaded at some point that did something similar. Google "Phet quantum tunelling". The java applet itself allows you to change the width and height of a rectangular potential barrier. Make the width small and the height large to approximate a delta potential. In your integral you have above you can't forget the time dependent factors. I'm pretty sure you have to include the e^(i*E(k)*t/h). I'll see if I can work through the fourier transforms later when I get home from work. $\endgroup$ – chuckstables Jul 27 '17 at 22:24
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    $\begingroup$ @Fly by Night - physicspages.com/2012/07/28/free-particle-gaussian-wave-packet. Here's a good website that has a bunch of common QM problems that you'd probably enjoy. The link here talks about constructing a gaussian wave packet for a free particle. I'd start with stuff like that before getting into constructing wave packets for particles in a potential. I tried to PM you the link but I couldn't figure out how (I know comments aren't for extended conversations- sorry :). $\endgroup$ – chuckstables Jul 27 '17 at 22:29
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    $\begingroup$ @FlybyNight A good reference is David Tannor's Quantum mechanics: a time-dependent perspective. $\endgroup$ – Emilio Pisanty Jul 27 '17 at 22:59

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