I have been learning various ways to solve TDSE and naturally, wavepacket motion seemed like a good test case to check the algorithms. Then, of course, I wanted to see one of the most interesting quantum phenomena - resonant tunneling.

I haven't been able to found any good animations about this phenomenon before, so I was quite surprised by what I saw. That's why I decided to ask the community what is happening here, and is my simulation giving physically correct results.

  1. I'm using absolute units, that is, nm and eV. Time is measured in $t/\hbar $, eV$^{-1}$, all the following simulations go from $t=0$ to $t/\hbar=67.5$ eV$^{-1}$.

  2. Electron has the usual mass, as if it's moving in vacuum.

  3. Wavepackets are gaussian, normalized with the same initial dispersion and initial group velocity determined by the input energy.

  4. I'm using transparent boundary conditions ($\partial_x \Psi(x_l) = -i k(E) \Psi(x_l)$, and $\partial_x \Psi(x_r) = i k(E) \Psi(x_r)$, where $x_l,x_r$ are left and right boundary).

The potential I'm using looks like this, and the three resonant energies I'm considering are listed at the top:

enter image description here

enter image description here

Now, I understand that since wavepacket has an energy spread, it's not going to experience resonant tunneling as a whole, only the part with energy close to the resonant one will pass through. On the other hand, even if the center energy is not at the resonance, some part is going to pass anyway.

That means that for the bottom resonances, which have very small broadening, most of the packet will be reflected, that's why I didn't even try to experiment with them.

(On the diagrams below the oscillatory line is the real part of the wavefunction, the smooth line - the absolute value, the color is the phase).

This is what I saw in the simulations, for example:

$$1) E_0 = \frac{\hbar^2 k_0^2}{2m} = 4.270245 ~eV$$

enter image description here

$$2) E_0 = \frac{\hbar^2 k_0^2}{2m} = 5.31959 ~eV$$

enter image description here

$$3) E_0 = \frac{\hbar^2 k_0^2}{2m} = 6.35275 ~eV$$

enter image description here

The cases 1 and 2 look almost the same, case 3 is clearly different, as in more of the wavepacket passes through.

  • In general, this makes sense to me, in that some part of the wavepacket passes through, some part doesn't, it depends apparently on the broadening of the resonance peak.

  • However I'm confused about the "resonant state" itself - that is, the standing wave between the barriers, that continues to oscillate and "radiate" something? What exactly is happening there? How much of "the particle" stays there and for how long?

  • And do my boundary conditions correctly describe the probability density inside the simulated region? It seems to me that since I start with a normalized state, then the integral of the probability density at time $t$ describes exactly "how much of the state" stays in the region.

If anyone would like to know more about the algorithm I'm using, I'll gladly elaborate.


1 Answer 1


There is a certain probability that the particle will tunnel through the fist barrier but then be reflected back and forth between the barriers. Each time it reaches a barrier there still is a certain small probability that it tunnels through it "radiating" some of the probability to that side.

And yes, if the boundaries are correctly implemented and numerical errors are not significant (small enough space between grid points and time steps), the integrated wave function gives you the probability of the particle still residing in the simulated area.

  • $\begingroup$ But this is a rather general picture. How is it related to the resonances of transmission probability? What is different when the energy is close to the resonance? $\endgroup$
    – Yuriy S
    Commented Dec 15, 2020 at 20:14
  • 1
    $\begingroup$ Well I don't think that this is something special about resonant tunneling. (Almost-)Resonant tunneling just causes the amount of probability that is trapped between the peaks to be relatively big and therefore easily observable. That doesn't mean that the same general thing isn't happening when you send a pulse around a non-resonant energy. With the amount of probability passing the first barrier then being smaller, it will just be harder to see on a plot/animation. $\endgroup$
    – paleonix
    Commented Dec 15, 2020 at 20:27
  • $\begingroup$ Thank you. I think you are right, as I observed almost the same picture between the resonances, but I thought it was dues to the energy distribution in the wavepacket. Probably, the resonance only occurs for a mythical coherent particle beam $\endgroup$
    – Yuriy S
    Commented Dec 15, 2020 at 20:29
  • $\begingroup$ You can try using less broad gaussian if you want to make sure it's the same :) $\endgroup$
    – paleonix
    Commented Dec 15, 2020 at 20:31
  • $\begingroup$ The problem is: less broad Gaussian immediately spreads around. The spread speed is inversely proportional to the space dispersion squared $\endgroup$
    – Yuriy S
    Commented Dec 15, 2020 at 20:33

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