Symplectic microgeometry, IV: Quantization

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.

The local conservation of a physical quantity whose distribution changes with time is mathematically described by the continuity equation. The corresponding time parameter, however, is defined with respect to an idealized classical clock. We consider what happens when this classical time is replaced by a non-relativistic quantum-mechanical description of the clock. From the clock-dependent Schr\"odinger equation (as analogue of the time-dependent Schr\"odinger equation) we derive a continuity equation, where, instead of a time-derivative, an operator occurs that depends on the flux (probability current) density of the clock. This clock-dependent continuity equation can be used to analyze the dynamics of a quantum system and to study degrees of freedom that may be used as internal clocks for an approximate description of the dynamics of the remaining degrees of freedom. As an illustration, we study a simple model for coupled electron-nuclear dynamics and interpret the nuclei as quantum clock for the electronic motion. We find that whenever the Born-Oppenheimer approximation is valid, the continuity equation shows that the nuclei are the only relevant clock for the electrons.

A suitable unified statistical formulation of quantum and classical mechanics in a *-algebraic setting leads us to conclude that information itself is noncommutative in quantum mechanics. Specifically we refer here to an observer's information regarding a physical system. This is seen as the main difference from classical mechanics, where an observer's information regarding a physical system obeys classical probability theory. Quantum mechanics is then viewed purely as a mathematical framework for the probabilistic description of noncommutative information, with the projection postulate being a noncommutative generalization of conditional probability. This view clarifies many problems surrounding the interpretation of quantum mechanics, particularly problems relating to the measuring process.

The review is based on the authro's papers since 1985 in which a new approach to the separation of variables (SoV) has being developed. It is argued that SoV, understood generally enough, could be the most universal tool to solve integrable models of the classical and quantum mechanics. It is shown that the standard constructuion of the action-angle variables from the poles of the Baker-Akhiezer function can be interpreted as a variant of SoV, and moreover, for many particular models it has a direct quantum counterpart. The list of the models discussed includes XXX and XYZ magnets, Gaudin model, Nonlinear Schrödinger equation, <it>S</it><it>L</it>(3)-invariant magnetic chain. New results for the 3-particle quantum Calogero-Moser system are reported.

Two types of prediction in Newtonian and quantum mechanics

Different approaches to the study of open linear freely-jointed (unhindered) threedimensional macromolecular chains at thermal equilibrium, based upon either Classical Hamiltonian Dynamics with constraints (namely, a formulation equivalent to Kramers' one) or Quantum Mechanics (with stiff harmonic springs), are reviewed and discussed in detail. Results from those approaches are compared to those arising from the standard Gaussian model. Fraenkel's approach, based upon Classical Mechanics (with stiff harmonic springs) and its comparison with the Gaussian model are also treated. It is argued that the quantum formulation is required in order that the treatment of the high-frequency vibrations be consistent. The role of the uncertainty principle is discussed, in support of the quantum-mechanical approach. At a later stage, internal rotations can be treated through Classical Statistical mechanics, under certain conditions. The quantum-Mechanics-based model appears to be consistent with the Gaussian model, while the one based upon constrained Hamiltonian Dynamics yields certain discrepancies with the latter. Classical and quantum-mechanical approaches to open linear freely-rotating (hindered) threedimensional macromolecular chains are also treated: the consistency of the quantum-mechanics-based model is displayed by deriving approximately the existence of a persistence length. It is argued that the possible advantages of the quantum-mechanical formulation should be expected not at large scales but at shorter ones. The specific topics to be treated are described in the summary of contents given below.

One of the most striking features of quantum mechanics is the profound effect exerted by measurements alone. Sophisticated quantum control is now available in several experimental systems, exposing discrepancies between quantum and classical mechanics whenever measurement induces disturbance of the interrogated system. In practice, such discrepancies may frequently be explained as the back-action required by quantum mechanics adding quantum noise to a classical signal. Here, we implement the "three-box" quantum game [Aharonov Y, et al. (1991) J Phys A Math Gen 24(10):2315-2328] by using state-of-the-art control and measurement of the nitrogen vacancy center in diamond. In this protocol, the back-action of quantum measurements adds no detectable disturbance to the classical description of the game. Quantum and classical mechanics then make contradictory predictions for the same experimental procedure; however, classical observers are unable to invoke measurement-induced disturbance to explain the discrepancy. We quantify the residual disturbance of our measurements and obtain data that rule out any classical model by ≳7.8 standard deviations, allowing us to exclude the property of macroscopic state definiteness from our system. Our experiment is then equivalent to the test of quantum noncontextuality [Kochen S, Specker E (1967) J Math Mech 17(1):59-87] that successfully addresses the measurement detectability loophole.

We discuss how the language of wave functions (state vectors) and associated non-commuting Hermitian operators naturally emerges from classical mechanics by applying the inverse Wigner-Weyl transform to the phase space probability distribution and observables. In this language, the Schr"odinger equation follows from the Liouville equation, with \hbarℏ now a free parameter. Classical stationary distributions can be represented as sums over stationary states with discrete (quantized) energies, where these states directly correspond to quantum eigenstates. Interestingly, it is now classical mechanics which allows for apparent negative probabilities to occupy eigenstates, dual to the negative probabilities in Wigner’s quasiprobability distribution. These negative probabilities are shown to disappear when allowing sufficient uncertainty in the classical distributions. We show that this correspondence is particularly pronounced for canonical Gibbs ensembles, where classical eigenstates satisfy an integral eigenvalue equation that reduces to the Schr"odinger equation in a saddle-point approximation controlled by the inverse temperature. We illustrate this correspondence by showing that some paradigmatic examples such as tunneling, band structures, Berry phases, Landau levels, level statistics and quantum eigenstates in chaotic potentials can be reproduced to a surprising precision from a classical Gibbs ensemble, without any reference to quantum mechanics and with all parameters (including \hbarℏ) on the order of unity.

The irreversible process of ‘wave packet collapse’ in quantum measurement contradicts the reversible characteristics of the Schrödinger equation and the entire quantum mechanics. This contradiction is actually the origin of measurement problems in quantum mechanics. How to resolve this contradiction is related to the understanding of some philosophical concepts. The Copenhagen School embraces the reversible/irreversible contradictions in quantum measurement through the principle of complementarity. Both von Neumann's measurement hypothesis and Everett's many world explanation adhere to the ‘universal position of the Schrödinger equation’, but due to their detachment from the thermodynamic mechanism analysis of quantum measurement processes, they fall into the dilemma of subjectivism or possible state materialization in ontology. Decoherence theory is a natural extension of dissipative structure theory in quantum mechanics, which helps to solve quantum measurement problems by combining the study of classical conversion mechanisms between reversibility and irreversibility in statistical mechanics. Due to the common statistical roots of the time arrow problem in classical mechanics and quantum mechanics, decoherence theory cannot be seen as a measurement theory that fully implements the ‘universal Schrödinger position’. Many contradictions in quantum mechanics may need to be cleverly resolved through various contradictions in classical physics.

We suggest a mathematical potential well with spherical symmetry and apply to the 1d Schr\"odinger equation. We use some well-known techniques as Stationary Perturbation Theory and WKB to gain insight into the solutions and compare them to each other. Finally, we solve the 1d Schr\"odinger equation using a numerical approach with the so-called Numerov technique for comparison. It can be a good exercise for undergrad students to grasp the above-cited techniques in a quantum mechanics course.

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.

With considerable interest I read the article “Bohr’s molecular model, a century later” by Anatoly Svidzinsky, Marlan Scully, and Dudley Herschbach (Physics Today, January 2014, page 33). It is always interesting to see how our current understanding sheds new light on the revolutionary scientific ideas of the past. I agree with the authors that although quantum mechanics is the real basis for atomic- and molecular-physics computations, the old Bohr model that treats electrons in atoms like tiny planets moving around the sun-nucleus is intuitively clear and very attractive.As I see it, the aim of the article is to show how to use Bohr’s approach to treat not only simple atoms but molecules. An important point in that approach is reconciliation of quantum mechanics with Bohr’s ideas. The authors claim that in infinite dimensions, quantum mechanics “morphs into classical mechanics.”I cannot say, however, that the infinite-dimension system is a clarifying model to describe physical or chemical objects. I do not see that the reference to chromodynamics (and to Edward Witten’s article in Physics Today, July 1980, page 38) is a clarifying one. Far from convincing are statements like “Hence the large-D limit, where 1/D → 0, is closer to the real world (1/D = 1/3) than is the oft-used D = 1 regime. Indeed, results obtained at large D usually resemble those for D = 3.” I confess it sounds too lacking in rigor to be convincing.But the large-D limit per se is not what bothers me. Of great concern is the assumption that the radial part of the D-dimensional Schrödinger equation in Hartree units looks like this:(1)where Z is the nuclear charge and l is the angular momentum.Equation (1) has no sense for D = 1. Indeed, in one dimension, a finite angular momentum requires infinite speed of a rotating particle. Therefore, for D = 1, the correct equation is(2)which does not follow from equation (1) at D = 1.But most important is that equation (1) implies that the Coulomb potential does not depend on D. That could have been considered correct before it became clear that the Coulomb law follows from Maxwell’s equations when one considers a field generated by a point-like electric charge. So to obtain the Coulomb law for a two-dimensional space, one has to consider Maxwell equations in a two-dimensional world. For a point-like charge, those equations lead to a 1/(ln r) instead of 1/r dependence. If one takes into account the D-dependence of the Coulomb potential, derivations performed in the Physics Today article become meaningless. Indeed, when using an unrealistic equation (1), how can one believe that it illuminates Bohr’s quite realistic and physical postulates?© 2014 American Institute of Physics.

The Weyl representation in classical and quantum mechanics

A quantum-mechanical transport equation corresponding to the classical Boltzmann equation is developed. Classical transport theory in the presence of a temperature gradient involves the phase space density, which has no meaning in quantum mechanics due to the complementarity of position and momentum. The superlattice representation allows the development of a quantum transport equation from the density matrix Schr\"odinger equation, through the introduction of irreversibility by standard methods. The quantum transport equation governing the electron occupation probability in the superlattice representation is derived for impurity scattering and for phonon scattering. This equation has the same field terms as the classical Boltzmann equation, but involves the discrete coarse-grained wave number and position of the superlattice representation. The scattering terms involve transitions between different superlattice states, and thus include scattering processes which move electrons from one superlattice cell to another. The solution of the classical equation is affected very little by these quantum effects, for all temperature gradients which might reasonably be encountered.

Quantum-Information Conservation. The Problem About ‘Hidden Variables’, or the ‘Conservation of Energy Conservation’ in Quantum Mechanics: A Historical Lesson for Future Discoveries