Quantum potential in time-dependent supersymmetric quantum mechanics

Theoretical physics seems to be in a kind of schizophrenic state. Many phenomena in the observable macroscopic world obey nonlinear evolution equations, whereas the microscopic world is governed by quantum mechanics, a fundamental theory that is supposedly linear. In order to combine these two worlds in a common formalism, at least one of them must sacrifice one of its dogmas. I claim that linearity in quantum mechanics is not as essential as it apparently seems since quantum mechanics can be reformulated in terms of nonlinear Riccati equations. In a first step, it will be shown where complex Riccati equations appear in time-dependent quantum mechanics and how they can be treated and compared with similar space-dependent Riccati equations in supersymmetric quantum mechanics. Furthermore, the time-independent Schrödinger equation can also be rewritten as a complex Riccati equation. Finally, it will be shown that (real and complex) Riccati equations also appear in many other fields of physics, like statistical thermodynamics and cosmology.

We propose the notion of a classical/quantum duality in the gravitational case (it can be extended to other interactions). By this one means exchanging Bohm's quantum potential for the classical potential VQ↔V in the stationary quantum Hamilton–Jacobi equation (QHJE) so that VQ+V=−V0 (ground state energy). Despite that the corresponding Schrödinger equations, and their solutions differ, their associated quantum Hamilton–Jacobi equation, and ground state energy remains the same. This is how the classical/quantum duality is implemented. In this scenario Bohm's quantum potential (which coincides with the attractive Newtonian potential) is now correlated to a classical repulsive gravitational potential (plus a constant). These results suggest that there might be a quantum origin to the classical repulsive gravitational behavior (of the accelerated expansion) of the universe which is based on this notion of classical/quantum duality. We hope that the notion of classical/quantum duality raised in this work in connection to the QHJE may cast further light into the deep interplay between gravity and quantum mechanics.

Several modifications of the Schr\"odinger equation that dynamically reduce the off-diagonal elements of the density operator in the position representation (localization) have received recent attention in the literature. Models of quantum open systems provide localization for the system of interest by considering the effects of an environment on the effective master equation that describes the dynamics of the reduced density operator for the system of interest. In contrast, localization is introduced as a fundamental modification to the Schr\"odinger equation in quantum mechanics with spontaneous localization (QMSL) [G. C. Ghirardi, A. Rhimini, and T. Weber, Phys. Rev. D 34, 470 (1986)]. We compare several environment models and discuss some of the anomalous features of the effective master equations noted by Ballentine [L. E. Ballentine, Phys. Rev. A 43, 9 (1991)] by examining the Ehrenfest relations for position, energy, and momentum. Insight is provided into the relative importance of various terms in the effective master equations for several environment models. We also briefly discuss the implications of similar anomalous features present in the QMSL master equation in light of our analysis of the open-system models.

We present an operator derivation of the continuity equation in quantum mechanics. We first identify the (local) divergence of the current density operator, then show how it immediately implies the continuity equation. We first consider the case of a free particle, then extend the treatment to the case of applied vector and scalar potentials.

Searching for the first well-documented attempts of introducing some kind of "quantization" into the description of nature inevitably leads to the ancient Greeks, in particular Plato and Pythagoras. The question of finding the so-called Pythagorean triples, i.e., right-angled triangles with integer length of all three sides, is, surprisingly, connected with complex nonlinear Riccati equations that occur in time-dependent quantum mechanics. The complex Riccati equation together with the usual Newtonian equation of the system, leads to a dynamical invariant with the dimension of an action. The relation between this invariant and a conserved "angular momentum" for the motion in the complex plane will be determined. The "Pythagorean quantization" shows similarities with the quantum Hall effect and leads to an interpretation of Sommerfeld's fine structure constant that involves another quantum of action, the "least Coulombic action" e2/c. Since natural evolution is characterized by irreversibility and dissipation, the question of how these aspects can be incorporated into a quantum mechanical description arises. Two effective approaches that also both possess a dynamical invariant (like the one mentioned above) will be discussed. One uses an explicitly time-dependent (linear) Hamiltonian, whereas the other leads to a nonlinear Schrödinger equation with complex logarithmic nonlinearity. Both approaches can be transformed into each other via a non-unitary transformation that involves Schrödinger's original definition of a (complex) action via the wave function.

The effect of sampling the electrostatic potential around a molecule on the quality of electrostatic potential derived charges is investigated. In addition, the effect of the number of expansion sites in a Distributed Multipole Analysis (DMA) on the quality of charges fitted to the DMA derived electrostatic potential is investigated. Sampling on constant electron density surfaces gives a better fit between the quantum mechanical potential and the potential derived from the fitted charges, compared to sampling on a van der Waals surface composed of intersecting spheres. The fit between the electrostatic potential derived from point charges and the quantum mechanical potential becomes poorer with increasing quality of the employed basis set. The inclusion of bond-centers into the calculations improves the fit between the Quantum Mechanical (QM) electrostatic potential and the DMA derived potential. The number of expansion sites needed for an accurate approximation of the QM electrostatic potential increases with increasing quality of the used basis set.

Two significant developments in the formulation of the equations of motion in the interaction representation (IR) are introduced in this paper. These developments have direct bearing on the efficient propagation in the interaction representation. The first of these developments involves the concept of sequential, or ‘‘nested’’ interaction transformations on the Hamiltonian and the wave function. Two such nested IRs are described. The first is a momentum- or ‘‘P-adapted’’ IR which, in addition to removing wave packet spreading, systematically maintains the average momentum 〈P〉 at zero. This facilitates a grid which not only has a minimal range, but whose points are maximally sparse. Evaluating the Hamiltonian operator in the P-adapted interaction representation involves only one simple algorithmic modification to the ordinary IR: replacing eiH0t/ℏV(R)e−iH0t /ℏ→eiH0t/ℏV(R+〈P〉t/m) eiH0t/ℏ, i.e., using the potential energy function which is downstream from the wave function by the classical propagation distance. A PR-adapted IR is also defined, which systematically maintains both the average position 〈R〉 and momentum 〈P〉 at zero. The PR-adapted IR also involves only one modification to the ordinary IR: replacing eiH0t/ℏV(R)e−iH0t /ℏ→eiH0t/ℏV(R+〈R〉 +〈P〉t/m)eiH0t/ℏ. The second formal development consists of the application of the operator equation eiH0t/ℏV(R)e−iH0t /ℏ=V(R+Pt/m) in conjunction with the nested IRs. This formal identity, which is essentially the Heisenberg representation for V(R), provides a convenient route to propagation in the interaction representation without reconstructing the Schrödinger wave function as an intermediate. The time propagation in all representations is performed using an iterative Lanczos reduction scheme combined with a second- order Magnus expansion. Numerical results for the exponential and the 1/R potential are presented, and illustrate that the new representations can easily lead to a savings of an order of magnitude in the size of the grid required for the propagation.

Ambiguity in the contact between laboratory instruments and equations of quantum mechanics is formulated in terms of responses of the instruments to commands transmitted to them by a Classical digital Process-control Computer (CPC); in this way instruments are distinguished from quantum-mechanical models (sets of equations) that specify what is desired of the instruments. Results include: (1) a formulation of quantum mechanics adapted to computer-controlled instruments; (2) a lower bound on the precision of unitary transforms required for quantum searching and a lower bound on sample size needed to show that instruments implement a desired model at that precision; (3) a lower bound on precision of timing required of a CPC in directing instruments; (4) a demonstration that guesswork is necessary in ratcheting up the precision of commands.

We review recent theoretical developments, which suggest that a set of shared principles underpin macroscopic quantum phenomena observed in high temperature super conducting materials, room temperature coherence in photosynthetic processes and the emergence of long range order in biological structures. These systems are driven by dissipative systems, which lead to fractal assembly and a fractal network of charges (with associated quantum potentials) at the molecular scale. At critical levels of charge density and fractal dimension, individual quantum potentials merge to form a charged induced macroscopic quantum potential, which act as a structuring force dictating long range order. Whilst the system is only partially coherent (i.e. only the bosonic fields are coherent), within these processes many of the phenomena associated with standard quantum theory are recovered, with macroscopic quantum potentials and associated forces having their equivalence in standard quantum mechanics. We establish a testable hypothesis that the development of structures analogous to those found in biological systems, which exhibit macroscopic quantum properties, should lead to increased critical temperatures in high temperature superconducting materials. If the theory is confirmed it opens up a new, systematic, ab initio approach to the structural development of these types of materials.

Complex adaptive learning is intelligent.It is adaptive, learns in feedback loops, and generates hidden patterns as many individuals, elements or particles interact in complex adaptive systems (CAS).CAS highlights adaptation in life and lifeless complex systems cutting across all traditional natural and social sciences disciplines.However, discovering a universal law in CAS and understanding the underlying mechanism of distribution formation, such as a non-Gauss distribution in complex quantum entanglement, remains highly challenging.Quantifying the uncertainty of CAS by probability waves, the authors explore the inherent logical relationship between Schrdinger's wave equation in quantum mechanics and Shi's trading volume-price wave equation in finance.Subsequently, the authors propose a non-localized wave equation in quantum mechanics if cumulative observable in a time interval represents momentum or momentum force in Skinner-Shi (reinforcement-frequency-interaction) coordinates.It supports the assumption that the invariance of interaction as a universal law exists in quantum mechanics and finance.The theory shows that quantum entanglement or adaptively interactive coherence is an interactively coherent state instead of a consequence of the superposition of coherent states.As a resource, quantum entanglement is non-separable, steerable, and energy-consumed.The entanglement state has opposite states subject to interaction conservation between the momentum and reversal forces.

The transition from reversible microscopic operator equations to irreversible equations for a deterministic density matrix is considered for examples of “simple systems”—the hydrogen atom or a free electron in an electromagnetic field. As a result of the transition, a system of a particle and field oscillators is replaced by a continuous medium. The Schrodinger equation for the deterministic wave function also describes the evolution of a continuum but without allowance for dissipative terms. In this sense, there is an analogy between the Schrodinger equation in quantum mechanics and Euler's equation in hydrodynamics. The smallest size of a “point” of a continuous medium is described by the classical electron radiusr e . It also determines the effective Thomson cross section for scattering of photons by free electrons. The lengthr e and the corresponding time interval τ e =r e /c play the role of “hidden parameters” in quantum mechanics. Two methods of calculating the effective Thomson cross section in terms of the extinction coefficient are considered. The first of them is based on the equation of motion of a free electron in a field with allowance for radiative friction. This equation leads to well-known difficulties. Moreover, the velocity fluctuations calculated on its basis lead to a contradiction with the second law of thermodynamics. The second method is based on the introduction of a constant friction coefficient Ń = τ e −1 , the presence of which reflects loss of information on smoothing over the volume of a “point” of the continuous medium. Such a method of calculation leads to the same expression for the effective cross section but makes it possible to avoid the difficulties with the second law of thermodynamics. In the derivation of quantum kinetic equations, the physically infinitesimally small scales are determined by the Compton length λC and the corresponding time interval. The introduction of these scales makes it possible to separate and eliminate small-scale fluctuations, the “collision integrals” being expressed in terms of the correlation functions of these fluctuations.

About the Different Approaches to Bohm's Quantum Potential in Non-Relativistic Quantum Mechanics

Within the context of supersymmetric quantum mechanics and its related hierarchies of integrable quantum Hamiltonians and potentials, a general programme is outlined and applied to its first two simplest illustrations. Going beyond the usual restriction of shape invariance for intertwined potentials, it is suggested to require a similar relation for Hamiltonians in the hierarchy separated by an arbitrary number of levels, N. By requiring further that these two Hamiltonians be in fact identical up to an overall shift in energy, a periodic structure is installed in the hierarchy which should allow for its resolution. Specific classes of orthogonal polynomials characteristic of such periodic hierarchies are thereby generated, while the methods of supersymmetric quantum mechanics then lead to generalised Rodrigues formulae and recursion relations for such polynomials. The approach also offers the practical prospect of quantum modelling through the engineering of quantum potentials from experimental energy spectra. In this paper, these ideas are presented and solved explicitly for the cases N = 1 and N = 2. The latter case is related to the generalised Laguerre polynomials, for which indeed new results are thereby obtained. In the context of dressing chains and deformed polynomial Heisenberg algebras, some partial results for N ⩾ 3 also exist in the literature, which should be relevant to a complete study of the N ⩾ 3 general periodic hierarchies.

Classical statistical average values are generally generalized to average values of quantum mechanics. It is discovered that quantum mechanics is a direct generalization of classical statistical mechanics, and we generally deduce both a new general continuous eigenvalue equation and a general discrete eigenvalue equation in quantum mechanics, and discover that a eigenvalue of quantum mechanics is just an extreme value of an operator in possibility distribution, the eigenvalue f is just classical observable quantity. A general classical statistical uncertain relation is further given, and the general classical statistical uncertain relation is generally generalized to the quantum uncertainty principle; the two lost conditions in classical uncertain relation and quantum uncertainty principle, respectively, are found. We generally expound the relations among the uncertainty principle, singularity and condensed matter stability, discover that the quantum uncertainty principle prevents the appearance of singularity of the electromagnetic potential between nucleus and electrons, and give the failure conditions of the quantum uncertainty principle. Finally, we discover that the classical limit of quantum mechanics is classical statistical mechanics, the classical statistical mechanics may further be degenerated to classical mechanics and we discover that merely stating that the classical limit of quantum mechanics is classical mechanics is a mistake. As application examples, we deduce both the Schrödinger equation and the state superposition principle, and deduce that there exists a decoherent factor from a general mathematical representation of the state superposition principle; the consistent difficulty between statistical interpretation of quantum mechanics and determinant property of classical mechanics is overcome.

In this dissertation, we search for a physical interpretation of the Aharonov-Bohm effect, one of the first and most famous cases of geometric phase in quantum mechanics. In order to do that, we use the bohmian formalism of quantum mechanics to formulate a precise condition (namely, the scale of spacial variation of the classical potential be greater than the characteristic dimension of the wave) for the classical limit and argue that the Aharonov-Bohm effect depends on the violation of this condition, which is possible due to the not simply connected topology of the phase space accessible to the eletron. Furthermore, we use the bohmiana mechanics to recover quantum mechanics from classical mechanics through a quasi-newtonian approach, showing that the quantum potential, which is que origem of non-classicality of quantum mechanics, is latent in the structure of classical mechanics. We present a thermodynamical interpretation for the quantum bohmian potential and which would be the machian origin of the quantum potential in the validity of the Mach Principle, in order to achieve a deeper compreheension of the foundations of quantum mechanics.