Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

CRC Press eBooks

Motion in a Magnetic Field

Lukong Cornelius Fai

Published Apr 19, 2022• 0 citations
Publisher

Abstract

The motion of a charged particle in a constant homogenous magnetic field leads to the motion of a harmonic oscillator. Examine first the motion of a charged particle in a constant magnetic field described by the Pauli Hamiltonian : 21.1 https://www.w3.org/1998/Math/MathML"> Η ^ = 1 2 m p ^ - e c Α → 2 + U - e ℏ 2 m c σ ^ , Η → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math21_1.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/>

DOI
10.1201/9781003273073-23

Related papers

Schrödinger Equation

Lukong Cornelius Fai

A linear harmonic oscillator is a system having 1 degree of freedom and executing in its direction a harmonic oscillation . We devote this heading to a study of an important physical system called the one-dimensional harmonic oscillator. We are concerned with a one-dimensional motion of a point mass that is attracted to a fixed center by a restoring force https://www.w3.org/1998/Math/MathML"> F x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math6_1.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> , which is proportional to the displacement https://www.w3.org/1998/Math/MathML"> x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math6_2.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> from that center: 6.1 https://www.w3.org/1998/Math/MathML"> F x = - m ω 2 x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math6_3.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/>

CRC Press eBooks•Apr 19, 2022•0 citations

One-Dimensional Motion General Principles

Lukong Cornelius Fai

Consider the properties of a harmonic oscillator: The motion is bounded: At infinity, the wave function https://www.w3.org/1998/Math/MathML"> Ψ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math5_1.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> tends to zero The energy levels are discrete These energy states are non-degenerate, i.e., any energy level corresponds to one eigenfunction The oscillatory theorem: The wave function https://www.w3.org/1998/Math/MathML"> Ψ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math5_2.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> of the https://www.w3.org/1998/Math/MathML"> n th https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math5_3.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> state has https://www.w3.org/1998/Math/MathML"> n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math5_4.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> zeros but not considering infinite far removed points

CRC Press eBooks•Apr 19, 2022•0 citations

Electron Spin in the Dirac Theory

Lukong Cornelius Fai

In non-relativistic quantum mechanics, the angular momentum, https://www.w3.org/1998/Math/MathML"> L ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math18_1.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> , is an integral of motion. The spin is added as a supplementary hypothesis. What should be the integral of motion in the Dirac theory for a particle described by the Hamiltonian, https://www.w3.org/1998/Math/MathML"> Η ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math18_2.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> ?: 18.1 https://www.w3.org/1998/Math/MathML"> Η ^ = c α ^ p ^ + β m c 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math18_3.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/>

CRC Press eBooks•Apr 19, 2022•0 citations

Propagator for a charged anisotropic oscillator in a constant magnetic field

N. Kokiantonis, K Castrigiano

The propagator for a charged, anisotropic harmonic oscillator in a constant magnetic field is computed. Making use of this an explicit expression is given for the path integral to an action with generalised memory recently considered in the literature.

Journal of Physics A Mathematical and General•Jan 11, 1985•20 citations

Schrödinger Equation

Lukong Cornelius Fai

In the previous chapter, we did not examine the change of a state with time https://www.w3.org/1998/Math/MathML"> t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_1.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . It has been assumed that we have been examining a state at a given moment https://www.w3.org/1998/Math/MathML"> t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_2.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . In the general case, the state of a system changes with time https://www.w3.org/1998/Math/MathML"> t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_3.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . For example, say the motion of an electron in a potential field, its wave function depends on time https://www.w3.org/1998/Math/MathML"> t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_4.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> : https://www.w3.org/1998/Math/MathML"> Ψ = Ψ q , t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_5.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . Such a dependence has to be described by a differential equation in which we may have https://www.w3.org/1998/Math/MathML"> ∂ Ψ ∂ t , ∂ 2 Ψ ∂ t 2 , ⋯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_6.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> .

CRC Press eBooks•Apr 19, 2022•0 citations

Exact Evaluation of the Propagator for a Charged Particle in a Constant Magnetic Field

Bin Kang Cheng

We are able to show that the propagator for a charged particle in a constant external magnetic field is related to that of a one-dimensional forced harmonic oscillator with memory, which can then be easily evaluated with the help of a gaussian integral.

Physica Scripta•Apr 1, 1984•7 citations
The motion of a charged particle in a constant homogenous magnetic field leads to the motion of a harmonic oscillator. Examine first the motion of a charged particle in a constant magnetic field described by the Pauli Hamiltonian : 21.1 https://www.w3.org/1998/Math/MathML"> Η ^ = 1 2 m p ^ - e c Α → 2 + U - e ℏ 2 m c σ ^ , Η → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math21_1.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/>
A linear harmonic oscillator is a system having 1 degree of freedom and executing in its direction a harmonic oscillation . We devote this heading to a study of an important physical system called the one-dimensional harmonic oscillator. We are concerned with a one-dimensional motion of a point mass that is attracted to a fixed center by a restoring force https://www.w3.org/1998/Math/MathML"> F x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math6_1.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> , which is proportional to the displacement https://www.w3.org/1998/Math/MathML"> x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math6_2.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> from that center: 6.1 https://www.w3.org/1998/Math/MathML"> F x = - m ω 2 x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math6_3.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/>
Consider the properties of a harmonic oscillator: The motion is bounded: At infinity, the wave function https://www.w3.org/1998/Math/MathML"> Ψ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math5_1.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> tends to zero The energy levels are discrete These energy states are non-degenerate, i.e., any energy level corresponds to one eigenfunction The oscillatory theorem: The wave function https://www.w3.org/1998/Math/MathML"> Ψ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math5_2.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> of the https://www.w3.org/1998/Math/MathML"> n th https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math5_3.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> state has https://www.w3.org/1998/Math/MathML"> n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math5_4.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> zeros but not considering infinite far removed points
In non-relativistic quantum mechanics, the angular momentum, https://www.w3.org/1998/Math/MathML"> L ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math18_1.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> , is an integral of motion. The spin is added as a supplementary hypothesis. What should be the integral of motion in the Dirac theory for a particle described by the Hamiltonian, https://www.w3.org/1998/Math/MathML"> Η ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math18_2.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> ?: 18.1 https://www.w3.org/1998/Math/MathML"> Η ^ = c α ^ p ^ + β m c 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math18_3.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/>
In the previous chapter, we did not examine the change of a state with time https://www.w3.org/1998/Math/MathML"> t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_1.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . It has been assumed that we have been examining a state at a given moment https://www.w3.org/1998/Math/MathML"> t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_2.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . In the general case, the state of a system changes with time https://www.w3.org/1998/Math/MathML"> t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_3.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . For example, say the motion of an electron in a potential field, its wave function depends on time https://www.w3.org/1998/Math/MathML"> t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_4.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> : https://www.w3.org/1998/Math/MathML"> Ψ = Ψ q , t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_5.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . Such a dependence has to be described by a differential equation in which we may have https://www.w3.org/1998/Math/MathML"> ∂ Ψ ∂ t , ∂ 2 Ψ ∂ t 2 , ⋯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_6.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> .