nLab Bohmian mechanics

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).

Related concepts

• interpretation of quantum mechanics

References

Named after:

• David Bohm, A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables. I, Phys. Rev. 85 (1952) 166 [doi:10.1103/PhysRev.85.166]

• David Bohm, A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables. II, Phys. Rev. 85 (1952) 180 [doi:10.1103/PhysRev.85.180]

Review:

• Antony Valentini: De Broglie-Bohm Quantum Mechanics, Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2409.01294]

See also:

• Wikipedia, De Broglie–Bohm theory

• Stanford Encyclopedia of Philosophy, Bohmian mechanics

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

A review of this relating to Bohmian mechanics:

• 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 (1992) 1175-1186

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) [doi:10.1126/sciadv.1501466]

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

category: physics