I Tunneling with Gaussian Wave Packet

[Tertius]

TL;DR Summary

I have a simulation built of an initial wave packet approaching a barrier and tunneling.

The goal is to have accurate 1D numerical results for tunneling probabilities through an arbitrary barrier without relying on analytic approximations such as WKB. If there is a more ideal approach to this, I am happy to change tactics. Time independent, for example, but I am not sure how to set up the boundaries at the edges so it doesn't become a bound system.

Most of the resources i've found detail way to solve for bound systems (oscillators, potential wells, etc), but I haven't found one that produces tunneling probabilities from a numerical method.

My initial simulation is a Gaussian wavepacket approaching a barrier, but I am finding that the numerical results are of course dependent on the initial location of the wavepacket. This is expected because the time evolution spreads out the wavepacket as it approaches the barrier.

I am attaching a snapshot of the simulation (both real and imaginary parts of $\psi$ are shown). I am computing the tunneling probability as the integrated probability after the barrier divided by the integrated probability before the barrier (computed at each time step, and taking the maximum).

I am concerned this approach is dependent on initial position. Is there a better general approach?

previously consulted resources:
https://arxiv.org/html/2403.13857v1#S3 https://www.reed.edu/physics/faculty/wheeler/documents/Quantum Mechanics/Miscellaneous Essays/Gaussian Wavepackets.pdf

 

[DrClaude]

My initial simulation is a Gaussian wavepacket approaching a barrier, but I am finding that the numerical results are of course dependent on the initial location of the wavepacket. This is expected because the time evolution spreads out the wavepacket as it approaches the barrier.

You should try a higher central momentum for the wave packet then. You want to be in a regime where the spreading of the wave packet is negligible (compared to $d\braket{x}/dt$) over the entire simulation.

I am attaching a snapshot of the simulation (both real and imaginary parts of $\psi$ are shown). I am computing the tunneling probability as the integrated probability after the barrier divided by the integrated probability before the barrier (computed at each time step, and taking the maximum).

In the conditions I mentioned above, you will get a splitting of the wave packet into separated reflected and transmitted parts, from which you can calculate the transmission probability, which will be independent of time once the interaction with the barrier is over (assuming that the wave packet never reaches the end of the grid). The maximum might not correspond to the long-time (converged) result.

Also, for square barriers you can get exact transmission probabilities for a plane wave (see, e.g., Sakurai's Modern Quantum Mechanics). The numerical result can of course differ as you are using a wave packet, but it is possible to make the wave packet narrow enough in momentum such that the two results are practically indistinguishable.

 

[Tertius]

That makes sense. And then, from what i've found in literature, it is common to have the spreading parameter $\sigma = \frac{h}{2 \lambda}$. That should, in combination with the momentum being high enough, give physically real and consistent results, I would hope.