[Paper Review] Bohmian Mechanics and the Meaning of the Wave Function

Detlef Dürr, Sheldon Goldstein+1 more|ArXiv.org|Dec 29, 1995

Quantum Mechanics and ApplicationsPhysics and Astronomy2 references137 citations

TL;DR

This paper argues that Bohmian mechanics provides a complete, deterministic description of quantum systems by combining particle positions (Q) with the wave function (ψ), resolving foundational issues like the measurement problem and Schrödinger's cat. It shows that the conditional wave function of a subsystem evolves according to the time-dependent Schrödinger equation even when the universal wave function is stationary, demonstrating how quantum dynamics emerges from a timeless, objective wave function in configuration space.

ABSTRACT

We outline how Bohmian mechanics works: how it deals with various issues in the foundations of quantum mechanics and how it is related to the usual quantum formalism. We then turn to some objections to Bohmian mechanics, for example the fact that in Bohmian mechanics there is no back action of particle configurations upon wave functions. These lead us to our main concern: a more careful consideration of the meaning of the wave function in quantum mechanics, as suggested by a Bohmian perspective. We propose that the reason, on the universal level, that there is no action of configurations upon wave functions, as there seems to be between all other elements of physical reality, is that the wave function of the universe is not an element of physical reality. We propose that the wave function belongs to an altogether different category of existence than that of substantive physical entities, and that its existence is nomological rather than material. We propose, in other words, that the wave function is a component of physical law rather than of the reality described by the law.

Motivation & Objective

• To address the foundational inadequacy in quantum mechanics regarding what the theory is fundamentally about—specifically, whether it describes particles, fields, or wave functions.
• To argue that Bohmian mechanics offers a more coherent ontology by treating particles (via their positions Q) as fundamental, with the wave function ψ as a guiding field.
• To clarify the physical meaning of the wave function, especially in subsystems, by showing how conditional wave functions evolve according to the Schrödinger equation despite a stationary universal wave function.
• To respond to objections—particularly from Shimony—by demonstrating that Bohmian mechanics naturally accounts for the appearance of wave function collapse and the use of the collapsed state in predictions.
• To show that the time-dependent Schrödinger equation for subsystems arises not as a fundamental law, but as an emergent feature of a universal, time-independent wave function in configuration space.

Proposed method

• The theory is formulated as a dynamical system where the state is (Q, ψ), with particle trajectories governed by the guidance equation: dQ/dt = Im(∇ψ/ψ)(Q).
• The universal wave function Ψ evolves unitarily via the Schrödinger equation: i∂ψ/∂t = Hψ, with H the Hamiltonian.
• The conditional wave function of a subsystem is defined as ψ_t(x) ∝ Ψ(x, Y(t)), where Y(t) is the actual configuration of the environment.
• The paper analyzes a model with two degrees of freedom (x and y), showing that when the y-system follows a classical-like trajectory, the x-system's conditional wave function evolves as if governed by a time-dependent Schrödinger equation.
• It derives the condition under which the conditional wave function satisfies i∂ψ_t/∂t = H_x ψ_t, by assuming the environment wave packets evolve unitarily as e^{-iH_y t}ϕ_0^α(y).
• The analysis shows that the effective dynamics of the subsystem emerge from the universal, time-independent Ψ when the environment's configuration follows a trajectory that tracks the center of a narrow wave packet.

Experimental results

Research questions

• RQ1 How can the time-dependent Schrödinger equation for a subsystem arise from a universal, time-independent wave function?
• RQ2 What is the physical meaning of the wave function in a quantum system, especially when it is not the complete description?
• RQ3 Can Bohmian mechanics resolve the measurement problem without invoking wave function collapse or observer effects?
• RQ4 Under what conditions does the conditional wave function of a subsystem obey the Schrödinger equation, even when the universal wave function is stationary?
• RQ5 How does decoherence relate to the emergence of effective dynamics in Bohmian mechanics?

Key findings

• The conditional wave function of a subsystem evolves according to the time-dependent Schrödinger equation, even when the universal wave function is time-independent, provided the environment's configuration follows a trajectory that tracks a narrow wave packet.
• When the environment's wave packets evolve unitarily as e^{-iH_y t}ϕ_0^α(y), the conditional wave function of the x-system satisfies i∂ψ_t/∂t = H_x ψ_t, demonstrating the emergence of standard quantum dynamics.
• The time evolution of the conditional wave function is projectively equivalent to a time-dependent state that satisfies the Schrödinger equation, even though the universal wave function is stationary.
• The result holds generally when the environment's wave packets are narrow and approximately disjoint, and their centers follow the actual particle trajectory Y(t), ensuring the conditional wave function remains well-defined and evolves unitarily.
• The emergence of the Schrödinger equation for subsystems is not contingent on the universal wave function being time-dependent, but rather on the dynamical tracking of the environment's configuration by its wave packet.
• Bohmian mechanics provides a realist, objective interpretation of the wave function as a field in configuration space, while also showing that the wave function of a subsystem can be time-dependent and physically meaningful, even if the universal wave function is not.