# nLab Bohmian mechanics

Contents

## Surveys, textbooks and lecture notes

#### Measure and probability theory

measure theory

probability theory

# Contents

## Idea

What is called Bohmian mechanics or de Broglie-Bohm theory is a rewriting of the Schrödinger equation of quantum mechanics in a way that makes it look – away from the zero locus of its solutions – like discribing a diffusion? process subject to an unusual (for ordinary diffusion) potential. There is a stochastic process modelling this diffusion (Nelson 66) and the point of view expressed by Bohmian mechanics is to regard this as a hidden variable theory of quantum mechanics.

Specifically, for the simple mechanical system of a particle of mass $m$ propagating on the real line $\mathbb{R}$ and subject to a potential $V \in C^\infty(\mathbb{R})$, the Schrödinger equation is the differential equation on complex-valued functions $\Psi \colon \mathbb{R}\times \mathbb{R} \to \mathbb{C}$ given by

$i \hbar \frac{\partial}{\partial t} \Psi = \frac{\hbar^2}{2m} \frac{\partial^2}{\partial^2 x} \Psi + V \Psi \,,$

where $\hbar$ denotes Planck's constant.

By the nature of complex numbers and by the discussion at phase and phase space in physics, it is natural to parameterize $\Psi$ – away from its zero locus – by a complex phase function

$S \;\colon\; \mathbb{R}\times \mathbb{R} \longrightarrow \mathbb{R}$

and an absolute value function $\sqrt{\rho}$

$\sqrt{\rho} \;\colon\; \mathbb{R}\times \mathbb{R} \longrightarrow \mathbb{R}$

which is positive, $\sqrt{\rho} \gt 0$, as

$\Psi \coloneqq \exp\left(\frac{i}{\hbar} S / \hbar\right) \sqrt{\rho} \,.$

Entering this Ansatz into the above Schrödinger equation, that complex equation becomes equivalent to the following two real equations:

$\frac{\partial}{\partial t} S = - \frac{1}{2m} \left(\frac{\partial}{\partial x}S\right)^2 - V + \frac{\hbar^2}{2m} \frac{1}{\sqrt{\rho}}\frac{\partial^2}{\partial^2 x} \sqrt{\rho}$

and

$\frac{\partial}{\partial t} \rho = - \frac{\partial}{\partial x} \left( \frac{1}{m} \left(\frac{\partial}{\partial x}S\right) \rho \right) \,.$

Now in this form one may notice a similarity of the form of these two equations with other equations from classical mechanics and statistical mechanics:

1. The first equation is similar to the Hamilton-Jacobi equation that expresses the classical action functional $S$ and the canonical momentum

$p \coloneqq \frac{\partial}{\partial x} S$

except that in addition to the ordinary potential energy $V$ there is an additional term

$Q \coloneq \frac{\hbar^2}{2m} \frac{1}{\sqrt{\rho}}\frac{\partial^2}{\partial^2 x} \sqrt{\rho}$

which is unlike what may appear in an ordinary Hamilton-Jacobi equation. The perspective of Bohmian mechanics is to regard this as a correction of quantum physics to classical Hamilton-Jacobi theory, it is then called the quantum potential. Notice that unlike ordinary potentials, this “quantum potential” is a function of the density that is subject to the potential. (Notice that this works only away from the zero locus of $\rho$.)

2. The second equation has the form of the continuity equation? of the flow expressed by $\frac{1}{m}p$.

The perspective of Bohmian mechanics is to regard this equivalent rewriting of the Schrödinger equation as providing a hidden variable theory formulation of quantum mechanics.

The bulk of the discussion of Bohmian mechanics in the literature revolves around philosophical implications of this perspective, e.g. (Stanford Enc. Philosph).

## References

The formulation of the exotic diffusion process described by the phase part of the Schrödinger equation as a stochastic process is due to

• Edward Nelson, Derivation of the Schrödinger equation from Newtonian mechanics, Phys. Rev. 150, 1079–1085, 1966

• Edward Nelson, Quantum Fluctuations, Princeton University Press,

Princeton. 1985

For a review of this relating to Bohmian mechanics see

• Guido Bacciagaluppi, A Conceptual Introduction to Nelson’s Mechanics (pdf)

The surreal trajectory problem is pointed out in

• B.-G. Englert, M. O. Scully, G. Süssmann, H. Walther, Surrealistic Bohm trajectories. Z. Naturforsch. 47a,1175 – 1186 (1992)

The following two articles offer solution to the surreal trajectory problem:

• D. H. Mahler et al. Experimental nonlocal and surreal Bohmian trajectories, Science Advances 2:2, e1501466 (2016) journal doi

• B. J. Hiley, R. Callaghan, O. Maroney, Quantum trajectories, real, surreal or an approximation to a deeper process? quant-ph/0010020

category: physics

Last revised on June 6, 2016 at 12:11:06. See the history of this page for a list of all contributions to it.