Computational applications of the many-interacting-worlds interpretation of quantum mechanics

While historically many quantum-mechanical simulations of molecular dynamics have relied on the Born-Oppenheimer approximation to separate electronic and nuclear behavior, recently a great deal of interest has arisen in quantum effects in nuclear dynamics as well. Due to the computational difficulty of solving the Schrödinger equation in full, these effects are often treated with approximate methods. In this paper, we present an algorithm to tackle these problems using an extension to the many-interacting-worlds approach to quantum mechanics. This technique uses a kernel function to rebuild the probability density, and therefore, in contrast with the approximation presented in the original paper, it can be naturally extended to $n$-dimensional systems. This opens up the possibility of performing quantum ground-state searches with steepest-descent methods, and it could potentially lead to real-time quantum molecular-dynamics simulations. The behavior of the algorithm is studied in different potentials and numbers of dimensions and compared both to the original approach and to exact Schrödinger equation solutions whenever possible.

Figure 1

Many-world potential for $N=2$ and for the cases of Gaussian and exponential kernels. It can be seen how the former features a minimum for the case of overlapping particles where the latter has a cusp. Units are arbitrary.

Figure 2

Logarithmic plots of the MIW energy error with an ideal distribution vs number of worlds for various potentials and dimensionalities. Ideal energies were computed with a matrix Numerov algorithm using grids of 200, 40, and 15 points of side, respectively, for 1D, 2D, and 3D. Circles represent the Gaussian kernel, triangles the exponential one, and squares the method from the original paper (only applicable to 1D). Filled dots represent uniformly distributed particles, empty ones the Monte Carlo distributed ones. Legend: (a), (b), and (c) are 1D potentials; (d), (e), and (f) are 2D; (g), (h), and (i) are 3D. By potential type, (a), (d), and (g) use the harm1 potential; (b), (e), and (h) use harm10; (c), (f), and (i) use lj1. The same labels apply to Figs. 3 and 4.

Figure 3

Energy convergence during the relaxation process for different potentials and dimensionalities. Continuous lines represent the Gaussian kernel, dot-dashed lines the exponential one, and dashed lines the method from [13] converged with damped Langevin MD. Empty circles and triangles represent, respectively, the Gaussian and exponential kernels converged with the BFGS algorithm. For the labels, refer to the caption in Fig. 2.

Figure 4

Ground-state density error RSS convergence during the relaxation process for different potentials and dimensionalities. Circles represent the Gaussian kernel, and triangles represent the exponential one. Full markers represent damped MD, whereas empty ones represent BFGS. It was not possible to compute the quantity for the original method as it does not provide a continuous approximation for the density. For the labels, refer to the caption in Fig. 2.

Figure 5

Top: Many-interacting-world trajectories at different temperatures and computed value of $\sqrt{\langle x^2\rangle }$ up to $2000\text{K}$ for a particle in a harmonic oscillator with $k=1\mathrm{eV}/{\mathring{\mathrm{A}}}^2$, using $N=30$ MIW worlds and an exponential kernel. Bottom: standard deviation for theoretical and computed densities with Gaussian and exponential kernels as a function of temperature.

Figure 6

Double harmonic well potential as described by Bell in [28]. The barrier height is $\mathrm{\Delta }E, x_0$ is the distance of the minimum from the barrier, and $a$ is the distance of the turning point, namely the point where the potential exceeds the zero point energy of the particle.

Figure 7

Jumping rates in a MIW simulation on a double harmonic well potential. The fitted parameters are shown with error bars (though most of them are so small as to be invisible) and overlapped with the Arrhenius and Bell models.