Bohm interpretation

The de Broglie–Bohm theory, also known as the pilot wave theory, Bohmian mechanics, Bohm's interpretation, and the causal interpretation, is an interpretation of quantum mechanics. It postulates that in addition to the wavefunction, an actual configuration of particles exists, even when unobserved. The evolution over time of the configuration of all particles is defined by a guiding equation. The evolution of the wave function over time is given by the Schrödinger equation. The theory is named after Louis de Broglie (1892–1987) and David Bohm (1917–1992).

The theory is deterministic^[1] and explicitly nonlocal: the velocity of any one particle depends on the value of the guiding equation, which depends on the configuration of all the particles under consideration.

Measurements are a particular case of quantum processes described by the theory—for which it yields the same quantum predictions generally associated with the Copenhagen interpretation. The theory does not have a "measurement problem", due to the fact that the particles have a definite configuration at all times. The Born rule in de Broglie–Bohm theory 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. There are several equivalent mathematical formulations of the theory.

Overview

De Broglie–Bohm theory is based on the following postulates:

There is a configuration $q$ of the universe, described by coordinates $q^{k}$ , which is an element of the configuration space $Q$ . The configuration space is different for different versions of pilot-wave theory. For example, this may be the space of positions $\mathbf {Q} _{k}$ of $N$ particles, or, in case of field theory, the space of field configurations $\phi (x)$ . The configuration evolves (for spin=0) according to the guiding equation $m_{k}{\frac {dq^{k}}{dt}}(t)=\hbar \nabla _{k}\operatorname {Im} \ln \psi (q,t)=\hbar \operatorname {Im} \left({\frac {\nabla _{k}\psi }{\psi }}\right)(q,t)={\frac {m_{k}\mathbf {j} _{k}}{\psi ^{*}\psi }}=\operatorname {Re} \left({\frac {\mathbf {\hat {P}} _{k}\Psi }{\Psi }}\right),$ where $\mathbf {j}$ is the probability current or probability flux, and $\mathbf {\hat {P}}$ is the momentum operator. Here, $\psi (q,t)$ is the standard complex-valued wavefunction known from quantum theory, which evolves according to Schrödinger's equation $i\hbar {\frac {\partial }{\partial t}}\psi (q,t)=-\sum _{i=1}^{N}{\frac {\hbar ^{2}}{2m_{i}}}\nabla _{i}^{2}\psi (q,t)+V(q)\psi (q,t).$

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