The de Broglie–Bohm theory, also known as the pilot wave theory, Bohmian mechanics, and the causal interpretation, is an interpretation of quantum mechanics that postulates that, in addition to the wave function, a particle possesses a definite position at all times, 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 interpretation is named after Louis de Broglie and David Bohm.
The interpretation is deterministic 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 as other interpretations of quantum mechanics. The interpretation 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 postulate. Rather, in this theory, the link between the probability density and the wave function has the status of a theorem, a result of a separate postulate, the "quantum equilibrium hypothesis", which is additional to the basic principles governing the wave function. There are several equivalent mathematical formulations of the interpretation.
Overview De Broglie–Bohm theory is based on the following postulates: There is a configuration of the universe, described by coordinates , which is an element of the configuration space . The configuration space is different for different versions of the pilot wave theory. For example, this may be the space of positions of particles, or, in case of field theory, the space of field configurations . The configuration evolves (for spin=0) according to the guiding equation where is the probability current or probability flux, and is the momentum operator. Here, is the standard complex-valued wavefunction from quantum theory, which evolves according to the Schrödinger equation This completes the specification of the theory for any quantum theory with Hamilton operator of type . The configuration is distributed according to at some moment of time , and this consequently holds for all times. Such a state is named quantum equilibrium. With quantum equilibrium, this theory agrees with the results of standard quantum mechanics.
Even though this latter relation is frequently presented as an axiom of the theory, Bohm presented it as derivable from statistical-mechanical arguments in the original papers of 1952. This argument was further supported by the work of Bohm in 1953 and was substantiated by Vigier and Bohm's paper of 1954, in which they introduced stochastic fluid fluctuations that drive a process of asymptotic relaxation from quantum non-equilibrium to quantum equilibrium (ρ →|ψ|2).Publications of D. Bohm in 1952 and 1953 and of J.-P. Vigier in 1954 as cited in p. 254.
Double-slit experiment
The double-slit experiment is an illustration of wave–particle duality. In it, a beam of particles (such as electrons) travels through a barrier that has two slits. If a detector screen is on the side beyond the barrier, the pattern of detected particles shows interference fringes characteristic of waves arriving at the screen from two sources (the two slits); however, the interference pattern is made up of individual dots corresponding to particles that had arrived on the screen. The system seems to exhibit the behaviour of both waves (interference patterns) and particles (dots on the screen).
If this experiment is modified so that one slit is closed, no interference pattern is observed. Thus, the state of both slits affects the final results. It can also be arranged to have a minimally invasive detector at one of the slits to detect which slit the particle went through. When that is done, the interference pattern disappears.
In de Broglie–Bohm theory, the wavefunction is defined at both slits, but each particle has a well-defined trajectory that passes through exactly one of the slits. The final position of the particle on the detector screen and the slit through which the particle passes is determined by the initial position of the particle. Such initial position is not knowable or controllable by the experimenter, so there is an appearance of randomness in the pattern of detection. In Bohm's 1952 papers he used the wavefunction to construct a quantum potential that, when included in Newton's equations, gave the trajectories of the particles streaming through the two slits. In effect the wavefunction interferes with itself and guides the particles by the quantum potential in such a way that the particles avoid the regions in which the interference is destructive and are attracted to the regions in which the interference is constructive, resulting in the interference pattern on the detector screen.
To explain the behavior when the particle is detected to go through one slit, one needs to appreciate the role of the conditional wavefunction and how it results in the collapse of the wavefunction; this is explained below. The basic idea is that the environment registering the detection effectively separates the two wave packets in configuration space.
Formulation Pilot wave theory The de Broglie–Bohm theory describes a pilot wave in a configuration space and trajectories of particles as in classical mechanics but defined by non-Newtonian mechanics. At every moment of time there exists not only a wavefunction, but also a well-defined configuration of the whole universe (i.e., the system as defined by the boundary conditions used in solving the Schrödinger equation).
The de Broglie–Bohm theory works on particle positions and trajectories like classical mechanics but the dynamics are different. In classical mechanics, the accelerations of the particles are imparted directly by forces, which exist in physical three-dimensional space. In de Broglie–Bohm theory, the quantum "field exerts a new kind of "quantum-mechanical" force". Bohm hypothesized that each particle has a "complex and subtle inner structure" that provides the capacity to react to the information provided by the wavefunction by the quantum potential.D. Bohm and B. Hiley: The undivided universe: An ontological interpretation of quantum theory, p. 37. Also, unlike in classical mechanics, physical properties (e.g., mass, charge) are spread out over the wavefunction in de Broglie–Bohm theory, not localized at the position of the particle.H. R. Brown, C. Dewdney and G. Horton: "Bohm particles and their detection in the light of neutron interferometry", Foundations of Physics, 1995, Volume 25, Number 2, pp. 329–347.J. Anandan, "The Quantum Measurement Problem and the Possible Role of the Gravitational Field", Foundations of Physics, March 1999, Volume 29, Issue 3, pp. 333–348.
The wavefunction itself, and not the particles, determines the dynamical evolution of the system: the particles do not act back onto the wave function. As Bohm and Hiley worded it, "the Schrödinger equation for the quantum field does not have sources, nor does it have any other way by which the field could be directly affected by the condition of the particles [...] the quantum theory can be understood completely in terms of the assumption that the quantum field has no sources or other forms of dependence on the particles". P. Holland considers this lack of reciprocal action of particles and wave function to be one "[a]mong the many nonclassical properties exhibited by this theory". Holland later called this a merely apparent lack of back reaction, due to the incompleteness of the description.
In what follows below, the setup for one particle moving in is given followed by the setup for N particles moving in 3 dimensions. In the first instance, configuration space and real space are the same, while in the second, real space is still , but configuration space becomes . While the particle positions themselves are in real space, the velocity field and wavefunction are on configuration space, which is how particles are entangled with each other in this theory.
Extensions to this theory include spin and more complicated configuration spaces.
We use variations of for particle positions, while represents the complex-valued wavefunction on configuration space.
Guiding equation For a spinless single particle moving in , the particle's velocity is
For many particles labeled for the -th particle their velocities are
The main fact to notice is that this velocity field depends on the actual positions of all of the particles in the universe. As explained below, in most experimental situations, the influence of all of those particles can be encapsulated into an effective wavefunction for a subsystem of the universe.
Schrödinger equation The one-particle Schrödinger equation governs the time evolution of a complex-valued wavefunction on . The equation represents a quantized version of the total energy of a classical system evolving under a real-valued potential function on :
For many particles, the equation is the same except that and are now on configuration space, :
This is the same wavefunction as in conventional quantum mechanics.
Relation to the Born rule
In Bohm's original papers, he discusses how de Broglie–Bohm theory results in the usual measurement results of quantum mechanics. The main idea is that this is true if the positions of the particles satisfy the statistical distribution given by . And that distribution is guaranteed to be true for all time by the guiding equation if the initial distribution of the particles satisfies .
For a given experiment, one can postulate this as being true and verify it experimentally. But, as argued by Dürr et al., one needs to argue that this distribution for subsystems is typical. The authors argue that , by virtue of its equivariance under the dynamical evolution of the system, is the appropriate measure of typicality for initial conditions of the positions of the particles. The authors then prove that the vast majority of possible initial configurations will give rise to statistics obeying the Born rule (i.e., ) for measurement outcomes. In summary, in a universe governed by the de Broglie–Bohm dynamics, Born rule behavior is typical.
The situation is thus analogous to the situation in classical statistical physics. A low-entropy initial condition will, with overwhelmingly high probability, evolve into a higher-entropy state: behavior consistent with the second law of thermodynamics is typical. There are anomalous initial conditions that would give rise to violations of the second law; however in the absence of some very detailed evidence supporting the realization of one of those conditions, it would be quite unreasonable to expect anything but the actually observed uniform increase of entropy. Similarly in the de Broglie–Bohm theory, there are anomalous initial conditions that would produce measurement statistics in violation of the Born rule (conflicting the predictions of standard quantum theory), but the typicality theorem shows that absent some specific reason to believe one of those special initial conditions was in fact realized, the Born rule behavior is what one should expect.
It is in this qualified sense that the Born rule is, for the de Broglie–Bohm theory, a theorem rather than (as in ordinary quantum theory) an additional postulate.
It can also be shown that a distribution of particles which is not distributed according to the Born rule (that is, a distribution "out of quantum equilibrium") and evolving under the de Broglie–Bohm dynamics is overwhelmingly likely to evolve dynamically into a state distributed as .. A video of the electron density in a 2D box evolving under this process is available here .