I’ve always been bothered by the way that scattering problems are introduced in introductory quantum mechanics courses. A typical problem is stated like this: given a particle moving toward a potential barrier, what is the probability that the particle tunnels through the barrier versus being reflected? Typically, these problems are approached by solving for the stationary states of the scattering potential Hamiltonian subject to certain constraints. Admittedly, the pedigree of this approach is impressive; it was employed by George Gamow in the first treatment of quantum tunneling. Yet at the same time, this approach contains troubling flaws. Most scattering problems have no stationary states – the eigenstates of their Hamiltonians are piecewise plane waves that are not normalizable. In order to treat them truly rigorously, a time-dependent approach must be employed.
To the rescue comes this excellent paper, that demonstrates how the standard plane-wave treatment is a limiting case of time-DEPENDENT Gaussian wave packet scattering. Whereas plane waves aren’t real (a given electron cannot be everywhere in space at once), Gaussian wave packets can be physically observed. The authors derive an improved formula for the tunneling probability that depends on the width of the initial wave packet.
I verified this result numerically using an FFT-based propagation method. You can play the scattering animation with the potential_step.py script found here. We start with a wave packet approaching a step potential:
After a certain amount of time, the wave packet will strike the barrier and split in two:
The two packets will then separate over time:
The following chart demonstrates the accuracy of the formula derived by McKagan et al.:
The main advantage of the wave packet approach is that the tunneling/reflection probabilities are obtained directly through standard “vanilla” quantum mechanics methods – just integrate the wave function in the region of interest! To me, this is much better than messing around with plane waves.