jyarger/Madelung-Equations

Computational software for solving the Madelung Equations. A project made for APM 598/MAT 494, Mathematics of Quantum Mechanics, at ASU

Madelung Equations

Computational software for solving the Madelung Equations. A project made for APM 598/MAT 494, Mathematics of Quantum Mechanics, at ASU

Authors: Eron Ristich, and Samarth Dev

Presented on December 7th, 2023, at Arizona State University. Presentation slides.

The Madelung Equations

The Madelung Equations are a set of equations proposed by Erwin Madelung in 1926 to describe a more physical interpretation of the Schrödinger Equation. Notably, Madelung's analysis resulted in a set of equations that are similar in nature to those of classical hydrodynamics, and are often referred to as the "hydrodynamic" interpretation of the Schrödinger Equation. The Madelung Equations are as follows:

$$ \begin{align} &\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{v}) = 0 \\ &\frac{\partial \vec{v}}{\partial t} + (\vec{v} \cdot \nabla)\vec{v} = -\frac{1}{m}\nabla (Q + V) \end{align} $$

where $\rho$ is the mass density, $\vec{v}$ is the velocity field, $Q$ is the Bohm quantum potential, $V$ is the potential energy, and $\psi$ is the wave function. $$\rho = m|\psi|^2$$ $$Q = -\frac{\hbar^2}{2m}\frac{\nabla^2 \sqrt{\rho}}{\sqrt{\rho}}$$

Notably, these equations take the form of the continuity equation and the Euler equation, respectively. The Madelung Equations are a set of nonlinear partial differential equations, and are difficult to solve analytically. However, they can be solved numerically using a variety of methods.

Derivation

The Madelung Equations can be derived from the Schrödinger Equation by first writing the wave function in polar form,

$$\psi = \sqrt{\frac{1}{m}\rho(\vec{x},t)}e^{iS/\hbar}$$

where $\rho$ is the mass density and $S$ is the quantum action. Substituting this into the Schrödinger Equation

$$i\hbar\frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2 \psi + V\psi$$

and separating the real and imaginary parts yields mass density conservation and the quantum Hamilton-Jacobi equation, respectively.

We define velocity as the gradient of the quantum action, $\vec{v} = \nabla S/m$, and mass density as the square of the wave function, $\rho = m|\psi|^2$. Substituting these into the quantum Hamilton-Jacobi equation yields the Madelung Equations.

Additionally, one can also find that probability current density $\vec{j}$ follows the same form as the classical hydrodynamics current density,

$$\vec{j} = \frac{1}{m} \rho \vec{v}$$

Numerical Discretization

We discretize the Madelung Equations using a technique common to classiscal computational fluid dynamics: smoothed particle hydrodynamics (SPH). One benefit of SPH over mesh-based methods is that it is a Lagrangian method, which is useful for adapting to complex or unbounded domains.

Our implementation follows that outlined by Mocz and Succi, 2015, adapted to two dimensions.

We choose a Gaussian kernel for the smoothing function as opposed to a cubic-spline. Although the cubic-spline has compact support (i.e. it is zero outside of a certain radius) and thus provides computational benefit over the Gaussian kernel, it is less accurate.

Installation

This project has been tested using Python 3.11.3 and Taichi 1.6.0, on Windows and Linux.

You can install the required packages using the following command:

pip install -r requirements.txt