Quantum/Classical Ensemble Correspondence
We study the correspondence between quantum wave packet dynamics and classical ensemble dynamics. The motivation is to find accurate methods to model quantum dynamics from classical trajectories and to study the instances where the correspondence holds to a good approximation, and when it fails. This is because classical simulations are in general much more computationally efficient than quantum simulations.
This could have applications such as simulating quantum dynamics in higher dimensions (e.g. molecular vibrational dynamics). Harmonic and barrier potentials were used to compare the quantum wave packet and classical ensemble dynamics.
The quantum wave packet dynamics were simulated using the split operator method. In contrast, the classical ensemble trajectories were simulated by sampling an initial position and momentum according to the probability densities in position and momentum space given by the wave function, then calculating its trajectory using Newtonian mechanics.
Figure 1: An instance where the quantum probability density agrees with the classical trajectories
Split-Operator Method for Quantum Dynamics
As the name suggests, the split-operator method is an approximation technique that involves splitting the time-evolution operator
into kinetic and potential parts by taking small steps in time (refer to [1] for a more detailed explanation):
Then we can approximate the time evolution of an initial wave function Ψ(x, 0) at t = 0 by repeatedly applying these operators:
This is implemented most easily by noting that the potential operator V is diagonal in the position representation, while the kinetic operator T is diagonal in the momentum representation. Therefore, after multiplying by the potential operator in position space, one uses the Fast Fourier Transform (FFT) to switch to momentum space, applies the kinetic operator (which is simply a multiplicative factor there), and then performs the inverse FFT back to position space:
where Ψ'' (x, t + ∆t ) is the wave function at time t = t + ∆t. Repeating this procedure N times will give you the time-propagated wave function at time t = N∆t.
Classical Ensemble Dynamics
Figure 2: A histogram of the sampled initial positions (blue) and the absolute value of the wave function squared (red)
Figure 3: A histogram of the sampled initial momenta (blue) and the absolute value of the wave function (in momentum space) squared (red)
Figure 4: The gaussian ground state wave function propagated using the split-operator method
Figure 5: A normalized histogram of all of the 50,000 classical trajectories propagated on a harmonic potential with initial conditions defined above.
Figure 6: A gaussian initial wave packet (wider than the ground state) propagated using the split-operator method
Figure 7: A normalized histogram of all of the 50,000 classical trajectories
