Quantum non-equilibrium is a concept within stochastic formulations of the De Broglie–Bohm theory of quantum physics.

In quantum mechanics, the Born rule states that the probability density of finding a system in a given state, when measured, is proportional to the square of the amplitude of the system's wavefunction at that state, and it constitutes one of the fundamental axioms of the theory.

This is not the case for the De Broglie–Bohm theory, where the Born rule is not a basic law. Rather, in this theory the link between the probability density and the wave function has the status of a hypothesis, called the quantum equilibrium hypothesis, which is additional to the basic principles governing the wave function, the dynamics of the quantum particles and the Schrödinger equation. (For mathematical details, refer to the derivation by Peter R. Holland.)

Accordingly, quantum non-equilibrium describes a state of affairs where the Born rule is not fulfilled; that is, the probability to find the particle in the differential volume ${\displaystyle d^{3}x}$ at time t is unequal to ${\displaystyle |\psi (\mathbf {x} ,t)|^{2}.}$

Recent advances in investigations into properties of quantum non-equilibrium states have been performed mainly by theoretical physicist Antony Valentini, and earlier steps in this direction were undertaken by David Bohm, Jean-Pierre Vigier, Basil Hiley and Peter R. Holland. The existence of quantum non-equilibrium states has not been verified experimentally; quantum non-equilibrium is so far a theoretical construct. The relevance of quantum non-equilibrium states to physics lies in the fact that they can lead to different predictions for results of experiments, depending on whether the De Broglie–Bohm theory in its stochastic form or the Copenhagen interpretation is assumed to describe reality. (The Copenhagen interpretation, which stipulates the Born rule a priori, does not foresee the existence of quantum non-equilibrium states at all.) That is, properties of quantum non-equilibrium can make certain classes of Bohmian theories falsifiable according to the criterion of Karl Popper.

In practice, when performing Bohmian mechanics computations in quantum chemistry, the quantum equilibrium hypothesis is simply considered to be fulfilled, in order to predict system behaviour and the outcome of measurements.

The causal interpretation of quantum mechanics has been set up by de Broglie and Bohm as a causal, deterministic model, and it was extended later by Bohm, Vigier, Hiley, Valentini and others to include stochastic properties.

Bohm and other physicists, including Valentini, view the Born rule linking ${\displaystyle R}$ to the probability density function ${\displaystyle \rho =R^{2}}$ as representing not a basic law, but rather as constituting a result of a system having reached quantum equilibrium during the course of the time development under the Schrödinger equation. It can be shown that, once an equilibrium has been reached, the system remains in such equilibrium over the course of its further evolution: this follows from the continuity equation associated with the Schrödinger evolution of ${\displaystyle \psi .}$ However, it is less straightforward to demonstrate whether and how such an equilibrium is reached in the first place.