Tipping time of a holonomic quantum cylinder

2011 ◽  
Vol 89 (9) ◽  
pp. 903-913
Author(s):  
Mark R.A. Shegelski ◽  
Jamie Sanchez-Fortun Stoker ◽  
Ian Kellett
Keyword(s):  
Wave Function ◽  
Time Scale ◽  
Numerical Solutions ◽  
Quantum Mechanical ◽  
Length Scale ◽  
Dimensionless Form ◽  
Order Unity ◽  
Holonomic Constraints ◽  
Initial Wave ◽  
Initial Wave Function

The classical and quantum mechanical Hamiltonians for a cylinder subject to holonomic constraints are derived. The quantum mechanical Hamiltonian is simplified and cast into a dimensionless form. The tipping time of a quantum mechanical cylinder subject to gravity is calculated. Numerical solutions for an appropriate initial wave function are obtained. We find that the tipping time is given by 〈t〉tip = t0C1 exp [C2(r/r0)9], where t0 is the time scale, C1 and C2 are constants of order unity, r is the radius of the cylinder, and r0 is the length scale for the tipping. We compare our results with those found in previous works.

scholarly journals Statistical Principles of Natural Philosophy

2021 ◽  
Author(s):  
Tao Guo
Keyword(s):  
Wave Function ◽  
Physical Model ◽  
Gravitational Force ◽  
Natural Philosophy ◽  
Relativistic Effects ◽  
Single Equation ◽  
Superstring Theory ◽  
Initial Wave ◽  
Self Consistent ◽  
Initial Wave Function

Currently, natural philosophy (Physics) lacks the most fundamental model and a complete set of self-consistent explanations. This article attempts to address several issues to fill in the gaps. Starting from the most basic philosophical paradoxes, I deduce a physical model (the natural philosophical outlook) to describe the laws governing the operation of the universe. Based on this model, a mathematical model is established to describe the generalized diffusion behavior of a moving particle swarm, and its simple verification is carried out. In this article, the gravitational force and relativistic effects are interpreted for the first time as a statistical effect of randomly moving particles. Thus, the gravitational force and special relativistic effects are integrated into a single equation (achieved by selecting an initial wave function with a specific norm when solving it), and the cause of stable particle formation is also revealed. The derived equation and the method of acquiring the initial wave function are fully self-consistent with the hypotheses stated in the physical model, thereby also proving the reliability of the physical model to some extent. Some of these ideas may have potential value as a basis for understanding the essence of quantum mechanics, relativity and superstring theory, as well as for gaining a further understanding of nature and the manufacture of quantum computers.

Theory of quantum arrival and spatial wave function collapse on the appearance of particle

2008 ◽  
Vol 465 (2101) ◽  
pp. 71-86 ◽  
Author(s):  
Eric A Galapon
Keyword(s):  
Wave Function ◽  
Coarse Graining ◽  
One Dimension ◽  
Time Of Arrival ◽  
Quantum Time ◽  
Initial Wave ◽  
First Time ◽  
Initial Wave Function ◽  
Arrival Point ◽  
Do So

The unexpected connection is unravelled between the collapse of the wave function on the appearance of particle and the quantum time-of-arrival problem in one dimension. To do so, a theory of quantum first time of arrival is developed in the interacting case for arbitrary arrival point in one dimension based on a self-adjoint and canonical coarse graining of a time-of-arrival operator that derives the classical time-of-arrival observable. The appearance of particle in quantum mechanics is then considered in the light of this theory. It is found that the appearance of particle arises as a combination of the collapse of the initial wave function into one of the eigenfunctions of the time-of-arrival operator, followed by a unitary Schrödinger evolution of the eigenfunction.

scholarly journals Normal typicality and von Neumann’s quantum ergodic theorem

2010 ◽  
Vol 466 (2123) ◽  
pp. 3203-3224 ◽  
Author(s):  
Sheldon Goldstein ◽  
Joel L. Lebowitz ◽  
Christian Mastrodonato ◽  
Roderich Tumulka ◽  
Nino Zanghì
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Density Matrix ◽  
Ergodic Theorem ◽  
Mathematical Structure ◽  
Von Neumann ◽  
Precise Formulation ◽  
Initial Wave ◽  
Large Systems ◽  
Initial Wave Function

We discuss the content and significance of John von Neumann’s quantum ergodic theorem (QET) of 1929, a strong result arising from the mere mathematical structure of quantum mechanics. The QET is a precise formulation of what we call normal typicality, i.e. the statement that, for typical large systems, every initial wave function ψ 0 from an energy shell is ‘normal’: it evolves in such a way that | ψ t 〉〈 ψ t | is, for most t , macroscopically equivalent to the micro-canonical density matrix. The QET has been mostly forgotten after it was criticized as a dynamically vacuous statement in several papers in the 1950s. However, we point out that this criticism does not apply to the actual QET, a correct statement of which does not appear in these papers, but to a different (indeed weaker) statement. Furthermore, we formulate a stronger statement of normal typicality, based on the observation that the bound on the deviations from the average specified by von Neumann is unnecessarily coarse and a much tighter (and more relevant) bound actually follows from his proof.

scholarly journals Consistency Proof for Multi-time Schrödinger Equations with Particle Creation and Ultraviolet Cut-Off

2021 ◽  
Author(s):  
Sascha Lill ◽  
Lukas Nickel ◽  
Roderich Tumulka
Keyword(s):  
Wave Function ◽  
Evolution Equations ◽  
Infinite System ◽  
Relativistic Particle ◽  
Particle Creation ◽  
Dense Subspace ◽  
Consistency Proof ◽  
Initial Wave ◽  
Time Variables ◽  
Initial Wave Function

AbstractFor multi-time wave functions, which naturally arise as the relativistic particle-position representation of the quantum state vector, the analog of the Schrödinger equation consists of several equations, one for each time variable. This leads to the question of how to prove the consistency of such a system of PDEs. The question becomes more difficult for theories with particle creation, as then different sectors of the wave function have different numbers of time variables. Petrat and Tumulka (2014) gave an example of such a model and a non-rigorous argument for its consistency. We give here a rigorous version of the argument after introducing an ultraviolet cut-off into the creation and annihilation terms of the multi-time evolution equations. These equations form an infinite system of coupled PDEs; they are based on the Dirac equation but are not fully relativistic (in part because of the cut-off). We prove the existence and uniqueness of a smooth solution to this system for every initial wave function from a certain class that corresponds to a dense subspace in the appropriate Hilbert space.

scholarly journals Statistical Principles of Natural Philosophy

2020 ◽  
Author(s):  
Tao Guo
Keyword(s):  
Wave Function ◽  
Physical Model ◽  
Gravitational Force ◽  
Natural Philosophy ◽  
Relativistic Effects ◽  
Single Equation ◽  
Superstring Theory ◽  
Initial Wave ◽  
Self Consistent ◽  
Initial Wave Function

Currently, natural philosophy (Physics) lacks the most fundamental model and a complete set of self-consistent explanations. This article attempts to discuss several issues related to this lack. Starting from the most basic philosophical paradoxes, I deduce a physical model (the natural philosophical outlook) to describe the laws governing the operation of the universe. Based on this model, a mathematical model is established to describe the generalized diffusion behavior of a moving particle swarm, and its simple verification is carried out. In this article, the gravitational force and relativistic effects are interpreted for the first time as a statistical effect of randomly moving particles. Thus, the gravitational force and special relativistic effects are integrated into a single equation (achieved by selecting an initial wave function with a specific norm when solving it), and the cause of stable particle formation is also revealed. The derived equation and the method of acquiring the initial wave function are fully self-consistent with the hypotheses stated in the physical model, thereby also proving the reliability of the physical model to some extent. Some of these ideas may have potential value as a basis for understanding the essence of quantum mechanics, relativity and superstring theory, as well as for gaining a further understanding of nature and the manufacture of quantum computers.

scholarly journals Initial wave function of the universe is arbitrary

2019 ◽  
Vol 34 (36) ◽  
pp. 1950295
Author(s):  
Ali Kaya
Keyword(s):  
Wave Function ◽  
Adjoint Operator ◽  
Scale Factor ◽  
Arbitrary Vector ◽  
Field System ◽  
Initial Wave ◽  
Condition Problem ◽  
The Universe ◽  
Initial Wave Function

We consider quantization of the gravity-scalar field system in the minisuperspace approximation. It turns out that in the gauge fixed deparametrized theory where the scale factor plays the role of time, the Hamiltonian can be uniquely defined without any ordering ambiguity as the square root of a self-adjoint operator. Moreover, the Hamiltonian degenerates to zero and the Schrödinger equation becomes well behaved as the scale factor vanishes. Therefore, there is no technical or physical obstruction for the initial wave function of the universe to be an arbitrary vector in the Hilbert space, which demonstrates the severeness of the initial condition problem in quantum cosmology.

scholarly journals Statistical Principles of Natural Philosophy

2020 ◽  
Author(s):  
Tao Guo
Keyword(s):  
Wave Function ◽  
Physical Model ◽  
Gravitational Force ◽  
Natural Philosophy ◽  
Relativistic Effects ◽  
Single Equation ◽  
Superstring Theory ◽  
Initial Wave ◽  
Self Consistent ◽  
Initial Wave Function

Currently, natural philosophy (Physics) lacks the most fundamental model and a complete set of self-consistent explanations. This article attempts to discuss several issues related to this lack. Starting from the most basic philosophical paradoxes, I deduce a physical model (the natural philosophical outlook) to describe the laws governing the operation of the universe. Based on this model, a mathematical model is established to describe the generalized diffusion behavior of a moving particle swarm, and its simple verification is carried out. In this article, the gravitational force and relativistic effects are interpreted for the first time as a statistical effect of randomly moving particles. Thus, the gravitational force and special relativistic effects are integrated into a single equation (achieved by selecting an initial wave function with a specific norm when solving it), and the cause of stable particle formation is also revealed. The derived equation and the method of acquiring the initial wave function are fully self-consistent with the hypotheses stated in the physical model, thereby also proving the reliability of the physical model to some extent. Some of these ideas may have potential value as a basis for understanding the essence of quantum mechanics, relativity and superstring theory, as well as for gaining a further understanding of nature and the manufacture of quantum computers.

WAVE-FUNCTION OF AN ASYMPTOTICALLY DE SITTER UNIVERSE

2007 ◽  
Vol 16 (09) ◽  
pp. 3014-3018 ◽  
Author(s):  
L. G. FERREIRA FILHO ◽  
J. ACACIO DE BARROS ◽  
E. V. CORRÊA SILVA ◽  
G. A. MONERAT ◽  
G. OLIVEIRA-NETO ◽  
...  
Keyword(s):  
Wave Function ◽  
Cosmological Constant ◽  
Scale Factor ◽  
De Sitter ◽  
Positive Cosmological Constant ◽  
Initial Wave ◽  
Frw Model ◽  
De Sitter Universe ◽  
Friedmann Robertson Walker ◽  
Initial Wave Function

In the present work, we quantize a closed Friedmann–Robertson–Walker (FRW) model in the presence of a positive cosmological constant and radiation. It gives rise to a Wheeler–DeWitt equation for the scale factor which has the form of a Schrödinger equation for a potential with a barrier. We solve it numerically and determine the evolution of an initial wave-function.

Atomic Imaging of Crystals using Large-Angle Electron Scattering in STEM

1990 ◽  
Vol 48 (1) ◽  
pp. 74-75
Author(s):  
D.E. Jesson ◽  
S. J. Pennycook
Keyword(s):  
Wave Function ◽  
Electron Scattering ◽  
Numerical Solutions ◽  
Large Angle ◽  
Real Space ◽  
High Angle ◽  
Analytical Treatment ◽  
Order Zero ◽  
Experimental Conditions ◽  
Ewald Sphere

It is well known that conventional atomic resolution electron microscopy is a coherent imaging process best interpreted in reciprocal space using contrast transfer function theory. This is because the equivalent real space interpretation involving a convolution between the exit face wave function and the instrumental response is difficult to visualize. Furthermore, the crystal wave function is not simply related to the projected crystal potential, except under a very restrictive set of experimental conditions, making image simulation an essential part of image interpretation. In this paper we present a different conceptual approach to the atomic imaging of crystals based on incoherent imaging theory. Using a real-space analysis of electron scattering to a high-angle annular detector, it is shown how the STEM imaging process can be partitioned into components parallel and perpendicular to the relevant low index zone-axis.It has become customary to describe STEM imaging using the analytical treatment developed by Cowley. However, the convenient assumption of a phase object (which neglects the curvature of the Ewald sphere) fails rapidly for large scattering angles, even in very thin crystals. Thus, to avoid unpredictive numerical solutions, it would seem more appropriate to apply pseudo-kinematic theory to the treatment of the weak high angle signal. Diffraction to medium order zero-layer reflections is most important compared with thermal diffuse scattering in very thin crystals (<5nm). The electron wave function ψ(R,z) at a depth z and transverse coordinate R due to a phase aberrated surface probe function P(R-RO) located at RO is then well described by the channeling approximation;

Operators and meaning of wave function

2016 ◽  
pp. 4039-4042
Author(s):  
Viliam Malcher
Keyword(s):  
Wave Function ◽  
Quantum Theory ◽  
State Vector ◽  
The State ◽  
Physical Significance ◽  
Central Problem ◽  
Quantum Mechanical ◽  
Mechanical Description ◽  
Quantum Mechanical Description ◽  
Interpretation Of Quantum Theory

The interpretation problems of quantum theory are considered. In the formalism of quantum theory the possible states of a system are described by a state vector. The state vector, which will be represented as |ψ> in Dirac notation, is the most general form of the quantum mechanical description. The central problem of the interpretation of quantum theory is to explain the physical significance of the |ψ>. In this paper we have shown that one of the best way to make of interpretation of wave function is to take the wave function as an operator.

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