75 Years of the Wavefunction: Complex-Dynamical Extension of the Original Wave Realism and the Universal Schroedinger Equation
Andrei P. Kirilyuk
备注：
34 pages, 27 eqs, 78 refs; Dedicated to the 75th Anniversary of the wavefunction and wave equation introduction by Erwin Schroedinger; Reviews a part of the author's book "Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge" (Kyiv, Naukova Dumka, 1997; in English), see physics/9806002; new text format, stylistic corrections in v3
Following Max Planck's hypothesis of quanta (quant-ph/0012069) and the matter wave idea of Louis de Broglie (quant-ph/9911107), Erwin Schroedinger proposed, at the beginning of 1926, the concept of wavefunction and wave equation for it. Though endowed with a realistic undular interpretation by its father, the wavefunction could not be considered as a real "matter wave" and has been provided with only abstract, formally probabilistic interpretation. In this paper we show how the resulting "mysteries" of usual theory are solved within the unreduced, dynamically multivalued description of the underlying, essentially nonlinear interaction process (quant-ph/9902015, quant-ph/9902016), without artificial modification of the Schroedinger equation. The latter is rigorously derived instead as universal expression of unreduced interaction complexity. Causal, totally realistic wavefunction is obtained as a dynamically probabilistic intermediate state of a simple system with interaction performing dynamically discrete transitions between its localised, incompatible "realisations" ("corpuscular" states). Causal wavefunction and Schroedinger equation are then extended to arbitrary level of world dynamics. We outline some applications of the obtained causal description, such as genuine quantum chaos (quant-ph/9511034-36) and realistic quantum devices (physics/0211071), and emphasize the basic difference of the proposed dynamically multivalued theory from dynamically single-valued imitations of causality and complexity. The causally complete wavefunction concept, representing the unified essence of unreduced (multivalued) complex dynamics, provides a clear distinctive feature of realistic science, absent in any its unitary imitation.
Efficient and accurate numerical propagation of the time dependent Schroedinger equation is a problem with applications across a wide range of physics. This paper develops an efficient, trivially parallelizeable method for relaxing a trial wavefunction toward a variationally optimum propagated wavefunction which minimizes the propagation error relative to a platonic wavefunction which obeys the time dependent Schroedinger equation exactly. This method is shown to be well suited for incorporation with multigrid methods, yielding rapid convergence to a minimum action solution even for Hamiltonia…
In the recent and very enjoyable paper (Paul Strange, "Semiclassical and Quantum Analysis of a Focussing Free Particle Hermite Wavefunction", arXiv:1309.6753[quant-ph]), Professor Strange has studied a particular solution of the free particle Schroedinger equation in which the time and space dependence are not separable. After recognizing the fact that "The Schroedinger equation has an identical mathematical form to the paraxial wave equation [...]", he claims to "describe and try to gain insight into an exotic, apparently accelerating solution of the free particle Schroedinger equation that i…
In the first paper of this series, I introduced a non-linear, Hamiltonian, generalization of Schroedinger's theory that blocks formation of macroscopic dispersion ("cats"). But that theory was entirely deterministic, and so the origin of random outcomes in experiments such as Stern-Gerlach or EPRB was left open. Here I propose that Schroedinger's wavefunction has a random component and demonstrate that such an improvised stochastic theory can violate Bell's inequality. Repeated measurements and the back-reaction on the microsystem are discussed in a toy example. Experiments that might falsify …
A simple real-space model for the electron wavefunction is suggested, based on a transverse wave with helicity, rotating at mc^2/h. The mapping of the real two-dimensional vector phasor to the complex plane permits this to satisfy the standard time-dependent Schroedinger equation. This model is extended to provide an intuitive physical picture of electron spin. Implications of this model are discussed.
We address the question whether Bohmian trajectories exist for all times. Bohmian trajectories are solutions of an ordinary differential equation involving a wavefunction obeying either the Schroedinger or the Dirac equation. Some trajectories may end in finite time, for example by running into a node of the wavefunction, where the law of motion is ill-defined. The aim is to show, under suitable assumptions on the initial wavefunction and the potential, global existence of almost all solutions. We provide a simpler and more transparent proof of the known global existence result for spinless Sc…
It has been proven theoretically for bosons with two-body repulsive interaction potentials in the dilute limit that the Gross-Pitaevskii equation provides the exact energy and density per particle as does the basic many-particle Schroedinger equation [Lieb and Seiringer, Phys. Rev. Lett. 88, 170409 (2002)]. Here, we investigate the overlap of the Gross-Pitaevskii and exact ground state wavefunctions. It is found that this overlap is always smaller than unity and may even vanish in spite of the fact that both wavefunctions provide the same energy and density per particle. Consequences are discu…
Here I show that a classical or quantum bit state plus one simple operation, an action, are sufficient ingredients to derive a quantum dynamical equation that rules the sequential changes of the state. Then, by assuming that a freely moving massive particle is the qubit carrier, it is found that both, the particle position in physical space and the qubit state, change in time according to the Pauli-Schroedinger equation. So, this approach suggests the following conjecture: because it carries one qubit of information the particle motion has its description enslaved by the very existence of the …
In this paper we consider the nonlinear one-dimensional time-dependent Schroedinger equation with a periodic potential and a local perturbation. In the limit of large periodic potential the time behavior of the wavefunction can be approximated, with a precise estimate of the remainder term, by means of the solution to the discrete nonlinear Schroedinger equation of the tight-binding model.
We consider a general class of discrete unitary dynamical models on the lattice. We show that generically such models give rise to a wavefunction satisfying a Schroedinger equation in the continuum limit, in any number of dimensions. There is a simple mathematical relationship between the mass of the Schroedinger particle and the eigenvalues of a unitary matrix describing the local evolution of the model. Second quantized versions of these unitary models can be defined, describing in the continuum limit the evolution of a nonrelativistic quantum many-body theory. An arbitrary potential is easi…
Quantum mechanics has about a dozen exactly solvable potentials. Normally, the time-independent Schroedinger equation for them is solved by using a generalized series solution for the bound states (using the Froebenius method) and then an analytic continuation for the continuum states (if present). In this work, we present an alternative way to solve these problems, based on the Laplace method. This technique uses a similar procedure for the bound states and for the continuum states. It was originally used by Schroedinger when he solved for the wavefunctions of hydrogen. Dirac advocated using …
The position operator (defined within the Schroedinger representation in the standard way) becomes meaningless when periodic boundary conditions are adopted for the wavefunction, as usual in condensed matter physics. We show how to define the position expectation value by means of a simple many-body operator acting on the wavefunction of the extended system. The relationships of the present findings to the Berry-phase theory of polarization are discussed.
An exactly solvable position-dependent mass Schrödinger equation in two dimensions, depicting a particle moving in a semi-infinite layer, is re-examined in the light of recent theories describing superintegrable two-dimensional systems with integrals of motion that are quadratic functions of the momenta. To get the energy spectrum a quadratic algebra approach is used together with a realization in terms of deformed parafermionic oscillator operators. In this process, the importance of supplementing algebraic considerations with a proper treatment of boundary conditions for selecting physical w…
This article examines the suggestion made in Ref. [EPL, 115 (2016) 60001] that a solution to a particle in an infinite spherical well model, if it is square-integrable, is a physically valid solution, even if at the precise location of the singularity there is no underlying physical cause, therefore, the divergence would have to be a nonlocal phenomenon caused by confining walls at a distance. In this work we examine this claim more carefully. By identifying the correct differential equation for a divergent square-integrable solution and rewriting it in the form of the Schroedinger equation, w…
The linearity of quantum mechanics leads, under the assumption that the wave function offers a complete description of reality, to grotesque situations famously known as Schroedinger's cat. Ways out are either adding elements of reality or replacing the linear evolution by a nonlinear one. Models of spontaneous wave function collapses took the latter path. The way such models are constructed leaves the question, whether such models are in some sense unique, i.e. whether the nonlinear equations replacing Schroedinger's equation, are uniquely determined as collapse equations. Various people work…
An alternative method is proposed for deriving the time dependent Schroedinger equation from the pictures of wave and matrix mechanics. The derivation is of a mixed classical quantum character, since time is treated as a classical variable, thus avoiding any controversy over its meaning in quantum mechanics. The derivation method proposed in this paper requires no ad hoc assumption and avoids going through a second-order differential equation that can be reduced to the well known time-dependent Schroedinger equation only postulating a complex wavefunction with an exponential time dependence, a…
Expressing the Schroedinger Lagrangian ${\cal L}$ in terms of the quantum wavefunction $ψ=\exp(S+{\rm i}I)$ yields the conserved Noether current ${\bf J}=\exp(2S)\nabla I$. When $ψ$ is a stationary state, the divergence of ${\bf J}$ vanishes. One can exchange $S$ with $I$ to obtain a new Lagrangian $\tilde{\cal L}$ and a new Noether current $\tilde{\bf J}=\exp(2I)\nabla S$, conserved under the equations of motion of $\tilde{\cal L}$. However this new current $\tilde{\bf J}$ is generally not conserved under the equations of motion of the original Lagrangian ${\cal L}$. We analyse the role playe…
Bassi and Ghirardi have developed a new theory to address the measurement problem based upon non-linear and stochastic modifications of the Schroedinger equation, which has been given the name Quantum Mechanical Spontaneous Localization or QMSL, with one of the emphasis dealing with reduction or wavefunction collapse within the nervous system. Analysis of this portion of their theory reveals that it faces several problems.
Generalized solutions of the standard gauge transformation equations are presented and discussed in physical terms. They go beyond the usual Dirac phase factors and they exhibit nonlocal quantal behavior, with the well-known Relativistic Causality of classical fields affecting directly the phases of wavefunctions in the Schroedinger Picture. These nonlocal phase behaviors, apparently overlooked in path-integral approaches, give a natural account of the dynamical nonlocality character of the various (even static) Aharonov-Bohm phenomena, while at the same time they seem to respect Causality. In…
We present a supersymmetric analysis for the d-dimensional Schroedinger equation with the generalized isotonic nonlinear-oscillator potential V(r)={B^2}/{r^{2}}+ω^{2} r^{2}+2g{(r^{2}-a^{2})}/{(r^{2}+a^{2})^{2}}, B\geq 0. We show that the eigenequation for this potential is exactly solvable provided g=2 and (ωa^2)^2 = B^2 +(\ell +(d-2)/2)^2. Under these conditions, we obtain explicit formulae for all the energies and normalized bound-state wavefunctions.
A generalized Euler equation of fluid dynamics is derived for describing many-body states of quantum mechanics. The Eulerian Eq. can be viewed as representing the interaction of two substates, where each substate has its own velocity and pressure fields. These field quantities are given by maps of the wavefunction. For one-body systems, the Eulerian Eq. can model either a fluid or particle description of quantum states. The generalized Euler Eq. is shown to be the gradient of an equation representing the total-energy of the two substates, having two energy fields. This total-energy Eq. is a ge…
Schroedinger's wave function shows many aspects of a state of incomplete knowledge or information ("bit"): (1) it is usually defined on a space of classical configurations, (2) its generic entanglement is, therefore, analogous to statistical correlations, and (3) it determines probabilities of measurement outcomes. Nonetheless, quantum superpositions (such as represented by a wave function) define individual physical states ("it"). This conceptual dilemma may have its origin in the conventional operational foundation of physical concepts, successful in classical physics, but inappropriate in q…
