WKB wave function for systems with many degrees of freedom: A unified view of solitons and pseudoparticles

1977 ◽  
Vol 16 (12) ◽  
pp. 3507-3514 ◽  
Author(s):  
J. -L. Gervais ◽  
B. Sakita
Keyword(s):  
Wave Function ◽  
Degrees Of Freedom ◽  
Unified View

scholarly journals Semi-classical approximations based on Bohmian mechanics

2020 ◽  
Vol 35 (14) ◽  
pp. 2050070 ◽  
Author(s):  
Ward Struyve
Keyword(s):  
Wave Function ◽  
Quantum Theory ◽  
Quantum Gravity ◽  
Quantum Electrodynamics ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Bohmian Mechanics ◽  
Expectation Values ◽  
Usual Approach ◽  
Classical Approximation

Semi-classical theories are approximations to quantum theory that treat some degrees of freedom classically and others quantum mechanically. In the usual approach, the quantum degrees of freedom are described by a wave function which evolves according to some Schrödinger equation with a Hamiltonian that depends on the classical degrees of freedom. The classical degrees of freedom satisfy classical equations that depend on the expectation values of quantum operators. In this paper, we study an alternative approach based on Bohmian mechanics. In Bohmian mechanics the quantum system is not only described by the wave function, but also with additional variables such as particle positions or fields. By letting the classical equations of motion depend on these variables, rather than the quantum expectation values, a semi-classical approximation is obtained that is closer to the exact quantum results than the usual approach. We discuss the Bohmian semi-classical approximation in various contexts, such as nonrelativistic quantum mechanics, quantum electrodynamics and quantum gravity. The main motivation comes from quantum gravity. The quest for a quantum theory for gravity is still going on. Therefore a semi-classical approach where gravity is treated classically may be an approximation that already captures some quantum gravitational aspects. The Bohmian semi-classical theories will be derived from the full Bohmian theories. In the case there are gauge symmetries, like in quantum electrodynamics or quantum gravity, special care is required. In order to derive a consistent semi-classical theory it will be necessary to isolate gauge-independent dependent degrees of freedom from gauge degrees of freedom and consider the approximation where some of the former are considered classical.

scholarly journals Quaternionic wave function

2019 ◽  
Vol 34 (02) ◽  
pp. 1950001 ◽  
Author(s):  
Pavel A. Bolokhov
Keyword(s):  
Wave Function ◽  
Quantum Electrodynamics ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Variational Calculus ◽  
Nonrelativistic Limit ◽  
Natural Choice ◽  
Complex Wave ◽  
Spinor Formalism ◽  
Function Language

We argue that quaternions form a natural language for the description of quantum-mechanical wave functions with spin. We use the quaternionic spinor formalism which is in one-to-one correspondence with the usual spinor language. No unphysical degrees of freedom are admitted, in contrast to the majority of literature on quaternions. In this paper, we first build a Dirac Lagrangian in the quaternionic form, derive the Dirac equation and take the nonrelativistic limit to find the Schrödinger’s equation. We show that the quaternionic formalism is a natural choice to start with, while in the transition to the noninteracting nonrelativistic limit, the quaternionic description effectively reduces to the regular complex wave function language. We provide an easy-to-use grammar for switching between the ordinary spinor language and the description in terms of quaternions. As an illustration of the broader range of the formalism, we also derive the Maxwell’s equation from the quaternionic Lagrangian of Quantum Electrodynamics. In order to derive the equations of motion, we develop the variational calculus appropriate for this formalism.

COLLECTIVE SPIN BY LINEARIZATION OF THE SCHRÖDINGER EQUATION FOR NUCLEAR COLLECTIVE MOTION

1988 ◽  
Vol 03 (09) ◽  
pp. 859-866 ◽  
Author(s):  
MARTIN GREINER ◽  
WERNER SCHEID ◽  
RICHARD HERRMANN
Keyword(s):  
Wave Function ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Degrees Of Freedom ◽  
Collective Motion ◽  
First Order ◽  
Energy And Momentum

The free Schrödinger equation for multipole degrees of freedom is linearized so that energy and momentum operators appear only in first order. As an example, we demonstrate the linearization procedure for quadrupole degrees of freedom. The wave function solving this equation carries a spin. We derive the operator of the collective spin and its eigenvalues depending on multipolarity.

INTERPRETATION AND PREDICTABILITY OF QUANTUM MECHANICS AND QUANTUM COSMOLOGY

1988 ◽  
Vol 03 (07) ◽  
pp. 645-651 ◽  
Author(s):  
SUMIO WADA
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Quantum Theory ◽  
Physical Meaning ◽  
Quantum Cosmology ◽  
Degrees Of Freedom ◽  
Interpretation Of Quantum Mechanics ◽  
Probabilistic Interpretation ◽  
The Universe ◽  
Classical Picture

A non-probabilistic interpretation of quantum mechanics asserts that we get a prediction only when a wave function has a peak. Taking this interpretation seriously, we discuss how to find a peak in the wave function of the universe, by using some minisuperspace models with homogeneous degrees of freedom and also a model with cosmological perturbations. Then we show how to recover our classical picture of the universe from the quantum theory, and comment on the physical meaning of the backreaction equation.

scholarly journals The cost of quantum locality

2021 ◽  
Vol 477 (2246) ◽  
pp. 20200602
Author(s):  
C. A. Bédard
Keyword(s):  
Wave Function ◽  
Degrees Of Freedom ◽  
Quantum Systems ◽  
Bell’S Theorem ◽  
Bell's Theorem ◽  
Global System ◽  
Small System ◽  
Local Description ◽  
Quantum Locality ◽  
The Cost

It has been more than 20 years since Deutsch and Hayden demonstrated that quantum systems can be completely described locally—notwithstanding Bell’s theorem. More recently, Raymond-Robichaud proposed two other approaches to the same conclusion. In this paper, all these means of describing quantum systems locally are proved formally equivalent. The cost of such descriptions is then quantified by the dimensionality of their underlining space. The number of degrees of freedom of a single qubit’s local description is shown to grow exponentially with the total number of qubits considered as a global system. This apparently unreasonable cost to describe such a small system in a large Universe is nonetheless shown to be expected. Finally, structures that supplement the universal wave function are investigated.

scholarly journals A minimalist pilot-wave model for quantum electrodynamics

2007 ◽  
Vol 463 (2088) ◽  
pp. 3115-3129 ◽  
Author(s):  
W Struyve ◽  
H Westman
Keyword(s):  
Wave Function ◽  
Electromagnetic Field ◽  
Quantum Theory ◽  
Quantum Electrodynamics ◽  
Degrees Of Freedom ◽  
Relativistic Quantum ◽  
Wave Model ◽  
Relativistic Quantum Theory ◽  
Pilot Wave ◽  
Free Electromagnetic Field

We present a way to construct a pilot-wave model for quantum electrodynamics. The idea is to introduce beables corresponding only to the bosonic and not to the fermionic degrees of freedom of the quantum state. We show that this is sufficient to reproduce the quantum predictions. The beables will be field beables corresponding to the electromagnetic field and will be introduced in a way similar to that of Bohm's model for the free electromagnetic field. Our approach is analogous to the situation in non-relativistic quantum theory, where Bell treated spin not as a beable but only as a property of the wave function. After presenting this model, we also discuss a simple way for introducing additional beables that represent the fermionic degrees of freedom.

scholarly journals Identification of Important Normal Modes in Nonadiabatic Dynamics Simulations by Coherence, Correlation, and Frequency Analyses

2019 ◽  
Author(s):  
Sebastian Mai ◽  
Leticia Gonzalez
Keyword(s):  
Wave Function ◽  
Degrees Of Freedom ◽  
Normal Modes ◽  
Dynamics Simulation ◽  
Nuclear Motion ◽  
Nonadiabatic Dynamics ◽  
Excitation Energies ◽  
Time Frequency ◽  
Dynamics Simulations ◽  
Low Dimensional

<div>Nonadiabatic dynamics simulations of molecular systems with a large number of nuclear degrees of freedom become increasingly feasible, but there is still a need to extract from such simulations a small number of most important modes of nuclear motion, for example to obtain general insight or to construct low-dimensional model potentials for further simulations. Standard techniques for this dimensionality reduction employ statistical methods that identify the modes that account for the largest variance in nuclear positions. However, large-amplitude motion is not necessarily a good proxy for the influence of a mode on the excited-state wave function evolution. Hence, here we report a number of analysis techniques aimed at extracting from nonadiabatic dynamics simulations the vibrational modes that are most strongly affected by the electronic excitation process and that most significantly affect the interaction of the electronic states. The first technique identifies coherent nuclear motion after excitation from the ratio between total variance and variance of the average trajectory. The second strategy employs linear regression to find normal modes that have a statistically significant effect on excitation energies, energy gaps, or wave function overlaps. The third approach uses time-frequency analysis to find normal modes where the vibrational frequencies change in the course of the dynamics simulation. All three techniques are applied to the case of surface hopping trajectories of [Re(CO)<sub>3</sub>(Im)(Phen)]<sup>+</sup> (Im=imidazole; Phen=1,10-phenanthroline), showing that in this transition metal complex the nonadiabatic dynamics is dominated by a small number of carbonyl and phenanthroline in-plane stretch modes.</div><div><br></div>

scholarly journals Coherence in the presence of absorption and heating in a molecule interferometer

2015 ◽  
Vol 6 (1) ◽  
Author(s):  
J. P. Cotter ◽  
S. Eibenberger ◽  
L. Mairhofer ◽  
X. Cheng ◽  
P. Asenbaum ◽  
...  
Keyword(s):  
Wave Function ◽  
Degrees Of Freedom ◽  
Center Of Mass ◽  
Dipole Interaction ◽  
Photon Absorption ◽  
Matter Wave ◽  
Complex Molecules ◽  
Particle Properties ◽  
Foundations Of Physics ◽  
Internal Degrees Of Freedom

Abstract Matter-wave interferometry can be used to probe the foundations of physics and to enable precise measurements of particle properties and fundamental constants. It relies on beam splitters that coherently divide the wave function. In atom interferometers, such elements are often realised using lasers by exploiting the dipole interaction or through photon absorption. It is intriguing to extend these ideas to complex molecules where the energy of an absorbed photon can rapidly be redistributed across many internal degrees of freedom. Here, we provide evidence that center-of-mass coherence can be maintained even when the internal energy and entropy of the interfering particle are substantially increased by absorption of photons from a standing light wave. Each photon correlates the molecular center-of-mass wave function with its internal temperature and splits it into a superposition with opposite momenta in addition to the beam-splitting action of the optical dipole potential.

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