Topic
Classical central-force problem
About: Classical central-force problem is a research topic. Over the lifetime, 152 publications have been published within this topic receiving 2469 citations.
Papers published on a yearly basis
Papers
TL;DR: In this article, the condition for simultaneous togetherness in a 3-body collision is expressed with the help of the 6×6 grand angular momentum tensor, Λ, whose components are Λij=(mimj)12xipj−(mjmi) 12xjpi,jΛij2 must be small.
Abstract: With short-range forces, initial and final states in a classical 3-body collision are straight-line trajectories into and out of a region where all three particles are close together at the same time. Using six coordinates, three describing the relative position of a pair of particles, and three the relative position of the third particle and the center of mass of the pair, the condition for simultaneous togetherness can be expressed with the help of the 6×6 grand angular momentum tensor, Λ, whose components are Λij=(mimj)12xipj−(mjmi)12xjpi. For a close 3-body collision Λ2=12Σi,jΛij2 must be small. Λ2 commutes with the ordinary angular momentum operators and with the kinetic energy; its eigenvalues are λ(λ+4)ℏ2, with integral λ, and its eigenfunctions hyperspherical harmonics. Initial and final 3-body states can be described quantally by the total energy E, Λ2, and a commuting set of ordinary angular momenta; this description has the same relation to a momentum representation as the ordinary angular momentum analysis has for a 2-body collision. A collision of (N+1) particles can be described by using a hierarchy of operators Λn2(2<~n<~N); their eigenvalues are λn(λn+3n−2)ℏ2.
373 citations
TL;DR: There is an additional nonconservative contribution to the scattering force arising in a light field with nonuniform helicity, which is shown to be proportional to the curl of the spin angular momentum of the light field.
Abstract: Light forces on small (Rayleigh) particles are usually described as the sum of two terms: the dipolar or gradient force and the scattering or radiation pressure force. The scattering force is traditionally considered proportional to the Poynting vector, which gives the direction and magnitude of the momentum flow. However, as we will show, there is an additional nonconservative contribution to the scattering force arising in a light field with nonuniform helicity. This force is shown to be proportional to the curl of the spin angular momentum of the light field. The relevance of the spin force is illustrated in the simple case of a 2D field geometry arising in the intersection region of two standing waves.
367 citations
Book•
12 Sep 2001
TL;DR: In this paper, the authors present a mathematical tool for quantum physics, including the Hilbert space and wave functions, to solve the problem of 3D problems in Cartesian Coordinates.
Abstract: Preface. 1. Origins of Quantum Physics. 1.1 Historical Note. 1.2 Particle Aspect of Radiation. 1.3 Wave Aspect of Particles. 1.4 Particles versus Waves. 1.5 Indeterministic Nature of the Microphysical World. 1.6 Atomic Transitions and Spectroscopy. 1.7 Quantization Rules. 1.8 Wave Packets. 1.9 Concluding Remarks. 1.10 Solved Problems. Exercises. 2. Mathematical Tools of Quantum Mechanics. 2.1 Introduction. 2.2 The Hilbert Space and Wave Functions. 2.3 Dirac Notation. 2.4 Operators. 2.5 Representation in Discrete Bases. 2.6 Representation in Continuous Bases. 2.7 Matrix and Wave Mechanics. 2.8 Concluding Remarks. 2.9 Solved Problems. Exercises. 3. Postulates of Quantum Mechanics. 3.1 Introduction. 3.2 The Basic Postulates of Quantum Mechanics. 3.3 The State of a System. 3.4 Observables and Operators. 3.5 Measurement in Quantum Mechanics. 3.6 Time Evolution of the System's State. 3.7 Symmetries and Conservation Laws. 3.8 Connecting Quantum to Classical Mechanics. 3.9 Solved Problems. Exercises. 4. One-Dimensional Problems. 4.1 Introduction. 4.2 Properties of One-Dimensional Motion. 4.3 The Free Particle: Continuous States. 4.4 The Potential Step. 4.5 The Potential Barrier and Well. 4.6 The Infinite Square Well Potential. 4.7 The Finite Square Well Potential. 4.8 The Harmonic Oscillator. 4.9 Numerical Solution of the Schrodinger Equation. 4.10 Solved Problems. Exercises. 5. Angular Momentum. 5.1 Introduction. 5.2 Orbital Angular Momentum. 5.3 General Formalism of Angular Momentum. 5.4 Matrix Representation of Angular Momentum. 5.5 Geometrical Representation of Angular Momentum. 5.6 Spin Angular Momentum. 5.7 Eigen functions of Orbital Angular Momentum. 5.8 Solved Problems. Exercises. 6. Three-Dimensional Problems. 6.1 Introduction. 6.2 3D Problems in Cartesian Coordinates. 6.3 3D Problems in Spherical Coordinates. 6.4 Concluding Remarks. 6.5 Solved Problems. Exercises. 7. Rotations and Addition of Angular Momenta. 7.1 Rotations in Classical Physics. 7.2 Rotations in Quantum Mechanics. 7.3 Addition of Angular Momenta. 7.4 Scalar, Vector and Tensor Operators. 7.5 Solved Problems. Exercises. 8. Identical Particles. 8.1 Many-Particle Systems. 8.2 Systems of Identical Particles. 8.3 The Pauli Exclusion Principle. 8.4 The Exclusion Principle and the Periodic Table. 8.5 Solved Problems. Exercises. 9. Approximation Methods for Stationary States. 9.1 Introduction. 9.2 Time-Independent Perturbation Theory. 9.3 The Variational Method. 9.4 The Wentzel "Kramers" Brillou in Method. 9.5 Concluding Remarks. 9.6 Solved Problems. Exercises. 10. Time-Dependent Perturbation Theory. 10.1 Introduction. 10.2 The Pictures of Quantum Mechanics. 10.3 Time-Dependent Perturbation Theory. 10.4 Adiabatic and Sudden Approximations. 10.5 Interaction of Atoms with Radiation. 10.6 Solved Problems. Exercises. 11. Scattering Theory. 11.1 Scattering and Cross Section. 11.2 Scattering Amplitude of Spinless Particles. 11.3 The Born Approximation. 11.4 Partial Wave Analysis. 11.5 Scattering of Identical Particles. 11.6 Solved Problems. Exercises. A. The Delta Function. A.1 One-Dimensional Delta Function. A.2 Three-Dimensional Delta Function. B. Angular Momentum in Spherical Coordinates. B.1 Derivation of Some General. B.2 Gradient and Laplacianin Spherical Coordinates. B.3 Angular Momentum in Spherical Coordinates. C. Computer Code for Solving the Schrodinger Equation. Index.
339 citations
Book•
1 Jan 1976
TL;DR: In this article, the authors present a mathematical model for the relative motion of a Particle and its relative motion in the plane of the ellipse of a rigid body, based on the concept of moments and products of inertia.
Abstract: Preface. 1. Fundamentals of Mechanics Review I. 2. Elements of Vector Algebra Review II. 3. Important Vector Quantities. 4. Equivalent Force Systems. 5. Equations of Equilibrium. 6. Introduction to Structural Mechanics. 7. Friction Forces. 8. Properties of Surfaces. 9. Moments and Products of Inertia. 10. Methods of Virtual Work and Stationary Potential Energy. 11. Kinematics of a Particle - Simple Relative Motion. 12. Particle Dynamics. 13. Energy Methods for Particles. 14. Methods of Momentum for Particles. 15. Kinematics of Rigid Bodies: Relative Motion. 16. Kinetics of Plane Motion of Rigid Bodies. 17. Energy and Impulse-Momentum Methods for Rigid Bodies. 18. Dynamics of General Rigid-Body Motion. 19. Vibrations. Appendix I. Integration Formulas. Appendix II. Computation of Principal Moments of Inertia. Appendix III. Additional Data For the Ellipse. Appendix IV. Proof that Infinitesimal Rotations Are Vectors.
135 citations
TL;DR: In this article, the traditional conservation laws for total charge, energy, linear and angular momentum, hold jointly in classical electron theory if and only if classical electron spin is included as dynamical degree of freedom.
45 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2018 | 1 |
| 2017 | 2 |
| 2016 | 6 |
| 2015 | 6 |
| 2014 | 3 |
| 2013 | 6 |