Particle on a polygon: Quantum Mechanics (original) (raw)
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Generalized centripetal force law and quantization of motion constrained on 2D surfaces
A B S T R A C T For a particle of mass μ moves on a 2D surface f x () = 0 embedded in 3D Euclidean space of coordinates x, there is an open and controversial problem whether the Dirac's canonical quantization scheme for the constrained motion allows for the geometric potential that has been experimentally confirmed. We note that the Dirac's scheme hypothesizes that the symmetries indicated by classical brackets among positions x and momenta p and
Classical and quantum mechanics of a particle on a rotating loop
American Journal of Physics, 2000
The toy model of a particle on a vertical rotating circle in the presence of uniform gravitational/ magnetic fields is explored in detail. After an analysis of the classical mechanics of the problem we then discuss the quantum mechanics from both exact and semi-classical standpoints. Exact solutions of the Schrödinger equation are obtained in some cases by diverse methods. Instantons, bounces are constructed and semi-classical, leading order tunneling amplitudes/decay rates are written down. We also investigate qualitatively the nature of small oscillations about the kink/bounce solutions. Finally, the connections of these toy examples with field theoretic and statistical mechanical models of relevance are pointed out.
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In this paper we review our results on the quantization of a rigid body. The fact that the configuration space is not simply connected yields two inequivalent quantizations. One of the quantizations allows us to recover classically double-valued wave functions as single valued sections of a non-trivial complex line bundle. This reopens the problem of a physical interpretation of these wave functions.
Classical and quantum study of the motion of a particle in a gravitational field
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The classical and the quantum mechanical description of a one-dimensional motion of a particle in the presence of a gravitational field is thoroughly discussed. The attention is centered on the evolution of classical and quantum mechanical position probability distribution function. The classical case has been compared with three different quantum cases: (a) a quantum stationary case, (b) a quantum non-stationary zero approximation case, where the wave packet has the shape of the first eigenfunction, and (c) a quantum non-stationary general case, where the wave packet is a superposition of stationary states.
Quantum mechanics of a generalised rigid body
2015
We consider the quantum version of Arnold's generalisation of a rigid body in classical mechanics. Thus, we quantise the motion on an arbitrary Lie group manifold of a particle whose classical trajectories correspond to the geodesics of any one-sided-invariant metric. We show how the derivation of the spectrum of energy eigenstates can be simplified by making use of automorphisms of the Lie algebra and (for groups of Type I) by methods of harmonic analysis. We show how the method can be extended to cosets, generalising the linear rigid rotor. As examples, we consider all connected and simply-connected Lie groups up to dimension 3. This includes the universal cover of the archetypical rigid body, along with a number of new exactly-solvable models. We also discuss a possible application to the topical problem of quantising a perfect fluid.
Canonical Quantization of Geometrized Mechanics
Journal of Physics: Conference Series
A particle in 3D space with certain potential will move in a curved trajectory like a comet in gravitational potential caused by the star. On the other hand, a free particle in curved space also moves according to geometry of that space. In this paper, the connection between potential energy and space metric will be discussed. So the formulation of classical mechanics in geometric terms can be found and the canonical quantization of it can be carried out. At the end of this paper, as an example, we will consider a particle under isotropic harmonic oscillator potential in two-dimensional sphere, carry out the canonical quantization, and then calculate the energies and their states.
On the Classical-Quantum Relation of Constants of Motion
Frontiers in Physics, 2018
Groenewold-Van Hove theorem suggest that is not always possible to transform classical observables into quantum observables (a process known as quantization) in a way that, for all Hamiltonians, the constants of motion are preserved. The latter is a strong shortcoming for the ultimate goal of quantization, as one would expect that the notion of "constants of motion" is independent of the chosen physical scheme. It has been recently developed an approach to quantization that instead of mapping every classical observable into a quantum observable, it focuses on mapping the constants of motion themselves. In this article we will discuss the relations between classical and quantum theory under the light of this new form of quantization. In particular, we will examine the mapping of a class of operators that generalizes angular momentum where quantization satisfies the usual desirable properties.
IJRASET, 2021
It is a well-known fact in physics that classical mechanics describes the macro-world, and quantum mechanics describes the atomic and sub-atomic world. However, principles of quantum mechanics, such as Heisenberg's Uncertainty Principle, can create visible real-life effects. One of the most commonly known of those effects is the stability problem, whereby a one-dimensional point base object in a gravity environment cannot remain stable beyond a time frame. This paper expands the stability question from 1-dimensional rod to 2-dimensional highly symmetrical structures, such as an even-sided polygon. Using principles of classical mechanics, and Heisenberg's uncertainty principle, a stability equation is derived. The stability problem is discussed both quantitatively as well as qualitatively. Using the graphical analysis of the result, the relation between stability time and the number of sides of polygon is determined. In an environment with gravity forces only existing, it is determined that stability increases with the number of sides of a polygon. Using the equation to find results for circles, it was found that a circle has the highest degree of stability. These results and the numerical calculation can be utilized for architectural purposes and high-precision experiments. The result is also helpful for minimizing the perception that quantum mechanical effects have no visible effects other than in the atomic, and subatomic world.
Physics of Atomic Nuclei, 2008
The quantum version of a nonlinear oscillator, previously analyzed at the classical level, is studied first in one dimension and then in two dimensions. This is a problem of quantization of a system with position-dependent mass of the form m = (1 + λx 2) −1 and with a λ-dependent nonpolynomial rational potential. The quantization procedure analyzes the existence of Killing vectors and makes use of an invariant measure. It is proved that this system can be considered as a model of the quantum harmonic oscillator on two-dimensional spaces of constant curvature.