Resonant tunneling for wavepackets, simulation - what exactly is happening here?

Question

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:

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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.

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(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:

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

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

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

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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.

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.