Quantum potential

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[3] and Carl Friedrich von Weizsäcker in 1935.

Hiley emphasised several aspects[10] that regard the quantum potential of a quantum particle: 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.

[13] 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.

[14] 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.

[16]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')".

[17] The Schrödinger wave function of a many-particle quantum system cannot be represented in ordinary three-dimensional space.

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.

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.

[24][25] In line with David Bohm's approach, Basil Hiley and mathematician Maurice de Gosson showed that the quantum potential can be seen as a consequence of a projection of an underlying structure, more specifically of a non-commutative algebraic structure, onto a subspace such as ordinary space (

-space, in similar manner as a Mercator projection inevitably results in a distortion in a geographical map.

Using this definition for the Fisher information, we can write:[37] Giovanni Salesi, Erasmo Recami and co-workers showed in 1998 that, in agreement with the König's theorem, the quantum potential can be identified with the kinetic energy of the internal motion ("zitterbewegung") associated with the spin of a spin-1/2 particle observed in a center-of-mass frame.

In this formulation, according to Esposito, quantum mechanics must necessarily be interpreted in probabilistic terms, for the reason that a system's initial motion condition cannot be exactly determined.

[42] James R. Bogan, in 2002, published the derivation of a reciprocal transformation from the Hamilton-Jacobi equation of classical mechanics to the time-dependent Schrödinger equation of quantum mechanics which arises from a gauge transformation representing spin, under the simple requirement of conservation of probability.

[43] B. Hiley and R. E. Callaghan re-interpret the role of the Bohm model and its notion of quantum potential in the framework of Clifford algebra, taking account of recent advances that include the work of David Hestenes on spacetime algebra.

, an element of the Clifford algebra which is isomorphic to the standard density matrix but independent of any specific representation.

Hiley and Callaghan distinguish bilinear invariants of a first kind, of which each stands for the expectation value of an element

In Riemann flat space, the Bohm potential is shown to equal the Weyl curvature.

According to Castro and Mahecha, in the relativistic case, the quantum potential (using the d'Alembert operator

[55] In relation to the Klein–Gordon equation for a particle with mass and charge, Peter R. Holland spoke in his book of 1993 of a "quantum potential-like term" that is proportional

He emphasized however that to give the Klein–Gordon theory a single-particle interpretation in terms of trajectories, as can be done for nonrelativistic Schrödinger quantum mechanics, would lead to unacceptable inconsistencies.

that are solutions to the Klein–Gordon or the Dirac equation cannot be interpreted as the probability amplitude for a particle to be found in a given volume

Holland pointed out that, while efforts have been made to determine a Hermitian position operator that would allow an interpretation of configuration space quantum field theory, in particular using the Newton–Wigner localization approach, but that no connection with possibilities for an empirical determination of position in terms of a relativistic measurement theory or for a trajectory interpretation has so far been established.

Yet according to Holland this does not mean that the trajectory concept is to be discarded from considerations of relativistic quantum mechanics.

as expression for the quantum potential, and he proposed a Lorentz-covariant formulation of the Bohmian interpretation of many-particle wave functions.

[61][62] Starting from the space representation of the field coordinate, a causal interpretation of the Schrödinger picture of relativistic quantum theory has been constructed.

real-valued functionals, can be shown[63] to lead to This has been called the superquantum potential by Bohm and his co-workers.

[66] In his article of 1952, providing an alternative interpretation of quantum mechanics, Bohm already spoke of a "quantum-mechanical" potential.

But the form of its activity is determined by the information content concerning its environment that is carried by the radar waves.

The quantum potential approach can be used to model quantum effects without requiring the Schrödinger equation to be explicitly solved, and it can be integrated in simulations, such as Monte Carlo simulations using the hydrodynamic and drift diffusion equations.