Shooting a classical particle
First, a video showing what happens when you try to shoot a classical particle when there's a potential barrier. For small initial kinetic energies, the particle bounces back. For large initial kinetic energies, the particle goes over the hump, first decelerating and then accelerating in the process.
Shooting a quantum particle
In the previous section we shot a classical particle at a potential barrier. Now let's shoot a quantum particle.
Whereas the behaviour of the classical particle is governed by Newton's Laws (where the external force providing the acceleration is given as minus the gradient of the potential), we allow our quantum particle to be governed by the Klein-Gordon equation.
- Mathematically, the Klein-Gordon equation is a partial differential equation, whereas Newton's laws form ordinary differential equations. A typical physical interpretation is that the state space of quantum particles are infinite dimensional, whereas the phase space of physics has finite dimensions.
- Note that physically the Klein-Gordon equation was designed to model a relativistic particle, while in the previous part we used the non-relativistic Newton's laws. In some ways it would've been better to model the quantum particle using Schrödinger's equation. I plead here however that (a) qualitatively there is not a big difference in terms of the simulated outcomes and (b) it is more convenient for me to use the Klein-Gordon model as I already have a finite-difference solver for hyperbolic PDEs coded in Python on my computer.
To model a particle, we set the initial data to be a moving wave packet, such that at the initial time the solution is strongly localized and satisfies $\partial_t u + \partial_x u = 0$. Absent the mass and potential energy terms in the Klein-Gordon equation (so under the evolution of the free wave equation), this wave packet will stay coherent and just translate to the right as time goes along. The addition of the mass term causes some small dispersion, but the mass is chosen small so that this is not a large effect. The main change to the evolution is the potential barrier, which you can see illustrated in the simulation.
The video shows 8 runs of the simulation with different initial data. Whereas in the classical picture the initial kinetic energy is captured by the initial speed at which the particle is moving, in the quantum/wave picture the kinetic energy is related to the central frequency of your wave packet. So each of the 8 runs have increasing frequency offset that represents increasing kinetic energy. The simulation has two plots, the top shows the square of the solution itself, which gives a good indication of where physically the wave packet is located. The bottom shows a normalized kinetic energy density (I have to include a normalization since the kinetic energy of the first and last particles differ roughly 10 fold).
One notices that in the first two runs, the kinetic energy is sufficiently small that the particle mostly bounces back to the left after hitting the potential.
For the third and fourth runs (frequency shift $\sqrt{2}$ and $\sqrt{3}$ respectively) we see that while a significant portion of the particle bounces back, a noticeable portion "tunnels through" the barrier: this caused by a combination of the quantum tunneling phenomenon and the wave packet form of the initial data.
The phenomenon of quantum tunneling manifests in that all non-zero energy waves will penetrate a finite potential barrier a little bit. But the amount of penetration decays to zero as the energy of the wave goes to zero: this is known as the semiclassical regime. In the semiclassical limit it is known that quantum mechanics converge toward classical mechanics, and so in the low-energy limit we expect our particle to behave like a classical particle and bounce off. So we see that naturally increasing the energy (frequency) of our wave packet we expect more of the tunneling to happen.
Further, observe that by shaping our data into a wave packet it necessarily contains some high frequency components (due to Heisenberg uncertainty principle); high frequency, and hence high energy components do not "see" the potential barrier. Even in the classical picture high energy particles would fly over the potential barrier. So for wave packets there will always be some (perhaps not noticeable due to the resolution of our computation) leakage of energy through the potential barrier. The quantum effect on these high energy waves is that they back-scatter. Whereas the classical high energy particles just fly directly over the barrier, a high energy quantum particle will leave some parts of itself behind the barrier always. We see this in the sixth and seventh runs of the simulation, where the particle mostly passes through the barrier, but a noticeable amount bounces off in the opposite direction.
In between during the fifth run, where the frequency shift is 2, we see that the barrier basically split the particle in two and send one half flying to the right and the other half flying to the left. Classically this is the turning point between particles that go over the bump and particles that bounces back, and would be the case (hard to show numerically!) where a classical particle comes in from afar with just enough energy that it comes to a half at the top of the potential barrier!
And further increasing the energy after the seventh run, we see in the final run a situation where only a negligible amount of the particle scatters backward with almost all of it passing through the barrier unchanged. One interesting thing to note however is that just like the case of the classical particle, the wave packet appears to "slow down" a tiny bit as it goes over the potential barrier.
Bouncing a quantum particle back and forth
In the two previous sections, I shot particles at a single potential barrier and looked at the result. What happens when we have more than one barrier? In the classical case the picture is easy to understand: a particle with insufficient energy to escape will be trapped in the local potential well forever, while a particle with sufficiently high energy will gain freedom and never come back. But what happens in the quantum case?
If the intuition we developed from scattering a quantum particle against a potential barrier, where we see that depending on the frequency (energy) of the particle, some portion gets transmitted and some portion gets reflected, is indeed correct, what we may expect to see is that the quantum particle bounces between the two barriers, each time losing some amplitude due to tunneling.
But we also saw that the higher frequency components of the quantum particle have higher transmission amplitudes. So we may expect that the high frequency components to decay more rapidly than the low frequency ones, so the frequency of the "left over" parts will continue to decay in time. This however, would be wrong, because we would be overlooking one simple fact: by the uncertainty principle again, very low frequency waves cannot be confined to a small physical region. So when we are faced with two potential barriers, the distance between them gives a characteristic frequency. Below this frequency (energy) it is actually not possible to fit a (half) wave between the barriers, and so the low frequency waves must have significant physical extent beyond the barriers, which means that large portions of these low frequency waves will just radiate away freely. Much above the characteristic frequency, however, the waves have large transmission coefficients and will not be confined.
So the net result is that we should expect for each double barrier a characteristic frequency at which the wave can remain "mostly" stuck between the two barriers, losing a little bit of amplitude at each bounce. This will look like a slowly, but exponentially, decaying standing wave. And I have some videos to show for that!
In the video we start with the same random initial data and evolve it under the linear wave equation with different potentials: the equations look like
\[ - \partial^2_{tt} u + \partial^2_{xx} u - V u = 0 \]
where $V$ is a non-negative potential taken in the form
\[ V(x) = a_1 \exp( - x^2 / b_1) - a_2 \exp( -x^2 / b_2) \]
which is a difference of two Gaussians. For the five waves shown the values of $a_1, b_1$ are the same throughout. The coefficients $a_2$ (taken to be $\leq a_1$) and $b_2$ (taken to be $< b_1$) increases from top to bottom, resulting in more and more-widely separated double barriers. Qualitatively we see, as we expected,
- The shallower and narrower the dip the faster the solution decays.
- The shallower and narrower the dip the higher the "characteristic frequency".
As an aside: the video shown above is generated using Python, in particular NumPy and MatPlotLib; the code took significantly longer to run (20+hours) than to write (not counting the HPDE solver I wrote before for a different project, coding and debugging this simulation took about 3 hours or less). On the other hand, this only uses one core of my quad-core machine, and leaves the computer responsive in the mean time for other things. Compare that to Auto-QCM: the last time I ran it to grade a stack of 400+ multiple choice exams it locked up all four cores of my desktop computer for almost an entire day.
As a further aside, this post is related somewhat to my MathOverflow question to which I have not received a satisfactory answer.
