Quantum potential

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. 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 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 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. 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 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. i \hbar \frac{\partial \psi}{\partial t} = \left( - \frac{\hbar^2}{2m} \nabla^2 +V \right)\psi \quad is re-written using the polar form for the wave function \psi = R \exp(i S / \hbar) with real-valued functions R and S , where R is the amplitude (absolute value) of the wave function \psi , and S/\hbar 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. \frac{\partial R}{\partial t} = -\frac{1}{2m} \left[ R \nabla^2 S + 2 \nabla R \cdot \nabla S \right], which, provided \rho = R^2 , can be interpreted as the continuity equation \partial \rho / \partial t + \nabla \cdot( \rho v) =0 for the probability density \rho and the velocity field v = \frac{1}{m}\nabla S Q(\mathbf{r_1},\mathbf{r_2},\,t) = - \frac{\hbar^2}{2m} \frac{(\nabla_1^2 + \nabla_2^2) R(\mathbf{r_1},\mathbf{r_2},\,t)}{R(\mathbf{r_1},\mathbf{r_2},\,t)} where \nabla_1^2 and \nabla_2^2 refer to particle 1 and particle 2 respectively. This expression generalizes in straightforward manner to n particles: Q(\mathbf{r_1},...,\mathbf{r_n},\,t) = -\frac{\hbar^2}{2 R(\mathbf{r_1},...,\mathbf{r_n},\,t) } \sum_{i=1}^{n} \frac{\nabla_i^2}{m_i} R(\mathbf{r_1},...,\mathbf{r_n},\,t) 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 \psi factorizes in the form \psi(\mathbf{r_1},\mathbf{r_2},\,t) = \psi_A(\mathbf{r_1},\,t) \psi_B(\mathbf{r_2},\,t) . Then it follows that also R factorizes, and the system's total quantum potential becomes the sum of the quantum potentials of the two particles. Q(\mathbf{r_1},\mathbf{r_2},\,t) = - \frac{\hbar^2}{2m} (\frac{\nabla_1^2 R_A(\mathbf{r_1},\,t)}{R_A(\mathbf{r_1},\,t)} + \frac{\nabla_2^2 R_B(\mathbf{r_2},\,t)}{R_B(\mathbf{r_2},\,t)}) = Q_A(\mathbf{r_1},\,t) + Q_B(\mathbf{r_2},\,t) In case the wave function is separable, that is, if \psi factorizes in the form \psi(\mathbf{r_1},\mathbf{r_2},\,t) = \psi_A(\mathbf{r_1},\,t) \psi_B(\mathbf{r_2},\,t) , the two one-particle systems behave independently. More generally, the quantum potential of an n -particle system with separable wave function is the sum of n quantum potentials, separating the system into n independent one-particle systems. \rho = R^2 \quad 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 \nabla^2 \sqrt \rho = \nabla \nabla \rho^{1/2} = \nabla \left(\frac{1}{2} \rho^{-1/2} \nabla \rho\right) = \frac{1}{2} \left[ \left(\nabla \rho^{-1/2}\right) \nabla \rho + \rho^{-1/2} \nabla^2 \rho \right] the quantum potential, expressed in terms of the probability density function, becomes: Q = - \frac{\hbar^2}{2m} \frac{\nabla^2 \sqrt{\rho}}{\sqrt{\rho}} = - \frac{\hbar^2}{4m} \left[ \frac{\nabla^2 \rho}{\rho} - \frac{1}{2} \frac{(\nabla \rho)^2}{\rho^2} \right] The quantum force F_Q = - \nabla Q , expressed in terms of the probability distribution, amounts to: F_Q = \frac{\hbar^2}{4m} \left[ \frac{\nabla (\nabla^2\rho)}{\rho} - \frac{ \nabla (\nabla \rho \cdot \nabla \rho) }{ 2\rho^2 } - \left( \frac{\nabla^2 \rho}{\rho} - \frac{ \nabla \rho \cdot \nabla \rho }{ \rho^2 } \right) \frac{\nabla\rho}{\rho} \right] It can be shown that the mean value of the quantum potential Q = - \hbar^2 \nabla^2 \sqrt{\rho} / (2m \sqrt{\rho}) is proportional to the probability density's Fisher information about the observable \hat{x} \mathcal{I} = \int \rho \cdot (\nabla \ln \rho)^2 \, d^3x = - \int \rho \nabla^2 (\ln \rho) \, d^3x. Using this definition for the Fisher information, we can write: \langle Q \rangle = \int \psi^* Q \psi \, d^3x = \int \rho Q \, d^3x = \frac{\hbar^2}{8m} \mathcal{I}. \mathbf V^2 = \frac{(\nabla \rho \land \mathbf s)^2} {(m \rho)^2} = \frac{(\nabla \rho)^2 \mathbf s^2 - (\nabla \rho \cdot \mathbf s)^2}{(m \rho)^2} from which the second term is shown to be of negligible size; then with | \mathbf s | = \hbar/2 it follows that | \mathbf V | = \frac{\hbar}{2} \frac{\nabla \rho}{m \rho} Salesi gave further details on this work in 2009. In 1999, Salvatore Esposito generalized their result from spin-1/2 particles to particles of arbitrary spin, confirming the interpretation of the quantum potential as a kinetic energy for an internal motion. Esposito showed that (using the notation \hbar =1) the quantum potential can be written as: Q = - \frac{1}{2} m \mathbf v_S^2 - \frac{1}{2} \nabla \cdot \mathbf v_S and that the causal interpretation of quantum mechanics can be reformulated in terms of a particle velocity \mathbf v = \mathbf v_B + \mathbf v_S \times \mathbf s where the "drift velocity" is \mathbf v_B = \frac {\nabla S}{m} and the "relative velocity" is \mathbf v_S \times \mathbf s , with \mathbf v_S = \frac {\nabla R^2}{2m R^2} and \mathbf s representing the spin direction of the particle. 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. Esposito explained that "the quantum effects present in the Schrödinger equation are due to the presence of a peculiar spatial direction associated with the particle that, assuming the isotropy of space, can be identified with the spin of the particle itself". Esposito generalized it from matter particles to gauge particles, in particular photons, for which he showed that, if modelled as \psi = (\mathbf E - i \mathbf B) / \sqrt 2 , with probability function \psi^* \cdot \psi = (\mathbf E^2 + \mathbf B^2)/2 , they can be understood in a quantum potential approach. 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. This spin-dependent transformation is a function of the quantum potential. 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. They show how, within a nested hierarchy of Clifford algebras C\ell_{i,j} , for each Clifford algebra an element of a minimal left ideal \Phi_L(\mathbf r, t) and an element of a right ideal representing its Clifford conjugation \Phi_R(\mathbf r, t) = \tilde{\Phi}_L(\mathbf r, t) can be constructed, and from it the Clifford density element (CDE) \rho_c(\mathbf r, t) = \Phi_L(\mathbf r, t) \tilde{\Phi}_L(\mathbf r, t) , an element of the Clifford algebra which is isomorphic to the standard density matrix but independent of any specific representation. On this basis, bilinear invariants can be formed which represent properties of the system. Hiley and Callaghan distinguish bilinear invariants of a first kind, of which each stands for the expectation value of an element B of the algebra which can be formed as {\rm Tr} B \rho_c , and bilinear invariants of a second kind which are constructed with derivatives and represent momentum and energy. Using these terms, they reconstruct the results of quantum mechanics without depending on a particular representation in terms of a wave function nor requiring reference to an external Hilbert space. Consistent with earlier results, the quantum potential of a non-relativistic particle with spin (Pauli particle) is shown to have an additional spin-dependent term, and the momentum of a relativistic particle with spin (Dirac particle) is shown to consist in a linear motion and a rotational part. The two dynamical equations governing the time evolution are re-interpreted as conservation equations. One of them stands for the conservation of energy; the other stands for the conservation of probability and of spin. The quantum potential plays the role of an internal energy which ensures the conservation of total energy. Carlo Castro and Jorge Mahecha derived the Schrödinger equation from the Hamilton-Jacobi equation in conjunction with the continuity equation, and showed that the properties of the relativistic Bohm quantum potential in terms of the ensemble density can be described by the Weyl properties of space. 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  \scriptstyle\Box and in the notation \hbar=1 ) takes the form Q = - \frac {1}{2m} \frac {\quad \Box \sqrt \rho}{\sqrt \rho} and the quantum force exerted by the relativistic quantum potential is shown to depend on the Weyl gauge potential and its derivatives. Furthermore, the relationship among Bohm's potential and the Weyl curvature in flat spacetime corresponds to a similar relationship among Fisher Information and Weyl geometry after introduction of a complex momentum. Diego L. Rapoport, on the other hand, associates the relativistic quantum potential with the metric scalar curvature (Riemann curvature). 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 \Box R/R . 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. For instance, wave functions \psi(\mathbf{x},t) 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 d^3 x at time t in accordance with the usual axioms of quantum mechanics, and similarly in the causal interpretation it cannot be interpreted as the probability for the particle to be in that volume at that time. 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. Hrvoje Nikolić derived Q = - (1/2m) \, \Box R/R as expression for the quantum potential, and he proposed a Lorentz-covariant formulation of the Bohmian interpretation of many-particle wave functions. He also developed a generalized relativistic-invariant probabilistic interpretation of quantum theory, in which |\psi|^2 is no longer a probability density in space but a probability density in space-time. Starting from the space representation of the field coordinate, a causal interpretation of the Schrödinger picture of relativistic quantum theory has been constructed. The Schrödinger picture for a neutral, spin 0, massless field \Psi \left[ \psi(\mathbf{x},t) \right] = R \left[ \psi(\mathbf{x},t) \right] e^{S \left[ \psi(\mathbf{x},t) \right]} , with R \left[ \psi(\mathbf{x},t) \right], S \left[ \psi(\mathbf{x},t) \right] real-valued functionals, can be shown to lead to Q \left[ \psi(\mathbf{x},t) \right] = - (1/2R) \int d^3 x \, \delta^2 R / \delta \psi^2 This has been called the superquantum potential by Bohm and his co-workers. Basil Hiley showed that the energy–momentum-relations in the Bohm model can be obtained directly from the energy–momentum tensor of quantum field theory and that the quantum potential is an energy term that is required for local energy–momentum conservation. He has also hinted that for particle with energies equal to or higher than the pair creation threshold, Bohm's model constitutes a many-particle theory that describes also pair creation and annihilation processes.

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