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De Broglie–Bohm theory
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De Broglie–Bohm theory
The de Broglie–Bohm theory is an interpretation of quantum mechanics which 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 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 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 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 theory.
De Broglie–Bohm theory is based on the following postulates:
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).
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.
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De Broglie–Bohm theory
The de Broglie–Bohm theory is an interpretation of quantum mechanics which 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 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 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 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 theory.
De Broglie–Bohm theory is based on the following postulates:
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).
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.