The quantum potential or quantum potentiality is a central concept of the de Broglie–Bohm formulation of quantum mechanics, introduced by David Bohm in 1952.
Initially presented under the name quantum-mechanical potential, subsequently quantum potential, it was later elaborated upon by Bohm and Basil Hiley in its interpretation as an information potential which acts on a quantum particle. It is also referred to as quantum potential energy, Bohm potential, quantum Bohm potential or Bohm quantum potential.
Quantum potential
In the framework of the de Broglie–Bohm theory, the quantum potential is a term within the Schrödinger equation which acts to guide the movement of quantum particles. The quantum potential approach introduced by Bohm (full text ) (full text ) provides a physically less fundamental exposition of the idea presented by Louis de Broglie: de Broglie had postulated in 1925 that the relativistic wave function defined on spacetime represents a pilot wave which guides a quantum particle, represented as an oscillating peak in the wave field, but he had subsequently abandoned his approach because he was unable to derive the guidance equation for the particle from a non-linear wave equation. The seminal articles of Bohm in 1952 introduced the quantum potential and included answers to the objections which had been raised against the pilot wave theory.
The Bohm quantum potential is closely linked with the results of other approaches, in particular relating to works of Erwin Madelung in 1927 and Carl Friedrich von Weizsäcker in 1935.
Building on the interpretation of the quantum theory introduced by Bohm in 1952, David Bohm and Basil Hiley in 1975 presented how the concept of a quantum potential leads to the notion of an "unbroken wholeness of the entire universe", proposing that the fundamental new quality introduced by quantum physics is nonlocality.D. Bohm, B. J. Hiley: On the intuitive understanding of nonlocality as implied by quantum theory, Foundations of Physics, Volume 5, Number 1, pp. 93-109, 1975, (abstract)
Relation to the Schrödinger equation
The Schrödinger equation
is re-written using the polar form for the wave function with real-valued functions and , where is the amplitude (absolute value) of the wave function , and its phase. This yields two equations: from the imaginary and real part of the Schrödinger equation follow the continuity equation and the quantum Hamilton–Jacobi equation respectively.David Bohm, Basil Hiley: The Undivided Universe: An Ontological Interpretation of Quantum Theory, Routledge, 1993, , therein Chapter 3.1. The main points of the causal interpretation, p. 22–23.
Continuity equation The imaginary part of the Schrödinger equation in polar form yields
which, provided , can be interpreted as the continuity equation for the probability density and the velocity field
Quantum Hamilton–Jacobi equation The real part of the Schrödinger equation in polar form yields a modified Hamilton–Jacobi equation
also referred to as quantum Hamilton–Jacobi equation.David Bohm, Basil Hiley: The Undivided Universe: An Ontological Interpretation of Quantum Theory, Routledge, 1993, , also as cited in: B. J. Hiley and R. E. Callaghan: Clifford Algebras and the Dirac-Bohm Quantum Hamilton-Jacobi Equation, Foundations of Physics, January 2012, Volume 42, Issue 1, pp 192-208 (published online 20 May 2011), (, 2010 preprint by B. Hiley) It differs from the classical Hamilton–Jacobi equation only by the term
This term , called quantum potential, thus depends on the curvature of the amplitude of the wave function.See for ex. Robert E. Wyatt, Eric R. Bittner: Quantum wave packet dynamics with trajectories: Implementation with adaptive Lagrangian grids of the amplitude of the wave function, Journal of Chemical Physics, vol. 113, no. 20, 22 November 2000, p. 8898 See also: Pilot wave#Mathematical formulation for a single particle
In the limit , the function is a solution of the (classical) Hamilton–Jacobi equation; therefore, the function is also called the Hamilton–Jacobi function, or action, extended to quantum physics.
Properties
Hiley emphasised several aspectsB. J. Hiley: Active Information and Teleportation, p. 7; appeared in: Epistemological and Experimental Perspectives on Quantum Physics, D. Greenberger et al. (eds.), pages 113-126, Kluwer, Netherlands, 1999 that regard the quantum potential of a quantum particle: it is derived mathematically from the real part of the Schrödinger equation under polar decomposition of the wave function,B.J. Hiley: From the Heisenberg picture to Bohm: A New Perspective on Active Information and it Relation to Shannon Information, pp. 2 and 5. Published in: A. Khrennikov (ed.): Proc. Conf. Quantum Theory: reconsideration of foundations, pp. 141–162, Vaxjö University Press, Sweden, 2002 is not derived from a Hamiltonian or other external source, and could be said to be involved in a self-organising process involving a basic underlying field; it does not change if is multiplied by a constant, as this term is also present in the denominator, so that is independent of the magnitude of and thus of field intensity; therefore, the quantum potential fulfils a precondition for nonlocality: it need not fall off as distance increases; it carries information about the whole experimental arrangement in which the particle finds itself.
In 1979, Hiley and his co-workers Philippidis and Dewdney presented a full calculation on the explanation of the two-slit experiment in terms of Bohmian trajectories that arise for each particle moving under the influence of the quantum potential, resulting in the well-known interference patterns.C. Philippidis, C. Dewdney, B. J. Hiley: , Il nuovo cimento B, vol. 52, no. 1, 1979, pp.15-28,
Also the shift of the interference pattern which occurs in presence of a magnetic field in the Aharonov–Bohm effect could be explained as arising from the quantum potential.C. Philippidis, D. Bohm, R. D. Kaye: The Aharonov-Bohm effect and the quantum potential, Il nuovo cimento B, vol. 71, no. 1, pp. 75-88, 1982,
Relation to the measurement process The collapse of the wave function of the Copenhagen interpretation of quantum theory is explained in the quantum potential approach by the demonstration that, after a measurement, "all the packets of the multi-dimensional wave function that do not correspond to the actual result of measurement have no effect on the particle" from then on.Basil J. Hiley: The role of the quantum potential. In: G. Tarozzi, Alwyn Van der Merwe: Open questions in quantum physics: invited papers on the foundations of microphysics, Springer, 1985, pages 237 ff., therein page 239 Bohm and Hiley pointed out that
Measurement then "involves a participatory transformation in which both the system under observation and the observing apparatus undergo a mutual participation so that the trajectories behave in a correlated manner, becoming correlated and separated into different, non-overlapping sets (which we call 'channels')".B. J. Hiley: The conceptual structure of the Bohm interpretation of quantum mechanics, In: , C. Montonen, K. Sunnarborg (eds.): Symposium on the Foundations of Modern Physics 1994 – 70 years of Matter Waves, Editions Frontières, pp. 99–118, , p. 106
Quantum potential of an n-particle system The Schrödinger wave function of a many-particle quantum system cannot be represented in ordinary three-dimensional space. Rather, it is represented in configuration space, with three dimensions per particle. A single point in configuration space thus represents the configuration of the entire n-particle system as a whole.
A two-particle wave function of identical particles of mass has the quantum potentialB. J. Hiley: Active Information and Teleportation, p. 10; appeared in: Epistemological and Experimental Perspectives on Quantum Physics, D. Greenberger et al. (eds.), pages 113-126, Kluwer, Netherlands, 1999
where and refer to particle 1 and particle 2 respectively. This expression generalizes in straightforward manner to particles:
In case the wave function of two or more particles is separable, then the system's total quantum potential becomes the sum of the quantum potentials of the two particles. Exact separability is extremely unphysical given that interactions between the system and its environment destroy the factorization; however, a wave function that is a superposition of several wave functions of approximately disjoint support will factorize approximately.
Derivation for a separable quantum system That the wave function is separable means that factorizes in the form . Then it follows that also factorizes, and the system's total quantum potential becomes the sum of the quantum potentials of the two particles.David Bohm, Basil Hiley: The Undivided Universe: An Ontological Interpretation of Quantum Theory, Routledge, 1993, , transferred to digital printing 2005, therein Chapter 4.1. The ontological interpretation of the many-body system, p. 59
In case the wave function is separable, that is, if factorizes in the form , the two one-particle systems behave independently. More generally, the quantum potential of an -particle system with separable wave function is the sum of quantum potentials, separating the system into independent one-particle systems.D. Bohm, B. J. Hiley, P. N. Kaloyerou: An ontological basis for the quantum theory, Physics Reports (Review section of Physics Letters), volume 144, number 6, pp. 323–348, 1987 (p. 351, eq. (12) <--page=31 p. 351 is not(!) a typo-->
Formulation in terms of probability density
Quantum potential in terms of the probability density function Bohm, as well as other physicists after him, have sought to provide evidence that the Born rule linking to the probability density function
can be understood, in a pilot wave formulation, as not representing a basic law, but rather a theorem (called quantum equilibrium hypothesis) which applies when a quantum equilibrium is reached during the course of the time development under the Schrödinger equation. With Born's rule, and straightforward application of the chain and product rules
the quantum potential, expressed in terms of the probability density function, becomes:See for example the Introduction section of: Fernando Ogiba: Phenomenological derivation of the Schrödinger equation , Progress in Physics (indicated date: October 2011, but retrieved online earlier: July 31, 2011)
Quantum force The quantum force , expressed in terms of the probability distribution, amounts to:Jeremy B. Maddox, Eric R. Bittner: Estimating Bohm's quantum force using Bayesian statistics , Journal of Chemical Physics, October 2003, vol. 119, no. 13, p. 6465–6474, therein p. 6472, eq.(38)
Formulation in configuration space and in momentum space, as the result of projections M. R. Brown and B. Hiley showed that, as alternative to its formulation terms of configuration space (-space), the quantum potential can also be formulated in terms of momentum space (-space).