Classical physics angular momentum

In the classical picture of an electron orbiting round the nucleus it would not surprise us to discover that the electron and the nucleus could each spin on its own axis, just like the earth and the moon, and that each has an angular momentum associated with spinning. Unfortunately, although quantum mechanical treatment gives rise to two new angular momenta, one associated with the electron and one with the nucleus, this simple physical... 

Our calculations have been successful in interpreting trends that are seen in the experimentally observed rates of electron ejection. However, until now, we have not had a clear physical picture of the energy and momentum (or angular momentum) balancing events that accompany such non BO processes. It is the purpose of this paper to enhance our understanding of these events by recasting the rate equations in ways that are more classical in nature (and hence hopefully more physically clear). This is done by... 

At the end of the nineteenth century classical physics assumed it had achieved a grand synthesis. The universe was thought of as comprising either matter or radiation as illustrated schematically in Fig. 2.1. The former consisted of point particles which were characterized by their energy E and momentum p and which behaved subject to Newton s laws of motion. The latter consisted of electromagnetic waves which were characterized by their angular frequency and wave vector and which satisfied Maxwell s recently discovered equations, ( = 2nv and — 2njX where v and X are the vibrational frequency... 

The physical significance of the particle-on-a-sphere wavefunctions is important in connection with the hydrogen atom. Comparing the energy formula (eqn 3.57) with the classical result (eqn 3.44) shows that the angular momentum is... 

Equation 4.7 is the Bohr postulate, that electrons can defy Maxwell s laws provided they occupy an orbit whose angular momentum (corresponding to an orbit of appropriate radius) satisfies Eq. 4.7. The Bohr postulate is not based on a whim, as most textbooks imply, but rather follows from (1) the Plank equation Eq. 4.3, AE = hv and (2) starting with an orbit of large radius such that the motion is essentially linear and classical physics applies, as no acceleration is involved, then extrapolating to small-radius orbits. The fading of quantum-mechanical equations into their classical analogues as macroscopic conditions are approached is called the correspondence principle [11]. 

We denote by G the set of all the experimentally observable quantities (called physical observables) which must be reproduced. Such quantities are, for instance, the collision energy, the quantum numbers defining the intramolecular state (vibrations and the principal quantum number of rotation), the total angular momentum etc... However, there are other dynamical variables which have a clear meaning in Classical Mechanics but correspond to no physical observable because of the Uncertainty Principle. We call them phase variables and denote them globally by g. The phase variables must be given particular values to obtain, at given G, a particular trajectory. Such variables are, for instance, the various intramolecular normal vibrational phases, the intermolecular orientation, the secondary rotation quantum numbers, the impact parameter, etc... Thus we look for relationships of the type qo = qq (G, g) and either qo = qo (G, g) or po = Po (G, g)... 

A time diagram of the electric field bursts (5.1.1) is shown in Fig. 5.3. A planar rotor perturbed by a sequence of delta kicks according to (5.1.1) is known as the kicked rotor. In order to familiarize ourselves with the physics of kicked systems, we focus in this section on the classical mechanics of the kicked rotor. If we denote by L the angular momentum of the rotor, the Hamiltonian function of the kicked rotor is given by... 

Auzinsli. M., Angular momenta dynamics in magnetic and electric field Classical and quantum approach. Canadian Journal of Physics, 75 (12) p. 853-872, (1997). R.N. Zare, Angular momentum (J. Wiley sons. New York, 1988), 280 pp. 

An inevitable consequence of de Broglie s standing-wave description of an electron in an orbit around the nucleus is that the position and momentum of a particle cannot both be known precisely and simultaneously. The momentum of the circular standing wave shown in Figure 4.18 is given exactly hj p = h/, but because the wave is spread uniformly around the circle, we cannot specify the angular position of the electron on the circle at all. We say the angular position is indeterminate because it has no definite value. This conclusion is in stark contrast with the principles of classical physics in which the positions and momenta are all known precisely and the trajectories of particles are well defined. How was this paradox resolved ... 

In addition to the quantum numbers n, l and m, which label its orbital, an electron is given an additional quantum number relating to an intrinsic property called spin, which is associated with an angular momentum about its own axis, and a magnetic moment. The rotation of planets about their axes is sometimes used as an analogy, but this can be misleading as spin is an essentially quantum phenomenon, which cannot be explained by classical physics. The direction of spin of an electron can take one of only two possible values, represented by the quantum number ms, which can have... 

Once the continuum hypothesis has been adopted, the usual macroscopic laws of classical continuum physics are invoked to provide a mathematical description of fluid motion and/or heat transfer in nonisothermal systems - namely, conservation of mass, conservation of linear and angular momentum (the basic principles of Newtonian mechanics), and conservation of energy (the first law of thermodynamics). Although the second law of thermodynamics does not contribute directly to the derivation of the governing equations, we shall see that it does provide constraints on the allowable forms for the so-called constitutive models that relate the velocity gradients in the fluid to the short-range forces that act across surfaces within the fluid. 

We thus have a simple model (the aufbau or building-up principle of Bohr [1] and Stoner [2]) which correctly predicts the periodic structure of Mendeleev s table of the elements. More precisely, one should state that Mendeleev s table is the experimental evidence which allows us to use an independent electron central field model and to associate each electron in a closed shell with a spherical harmonic of given n and i, because there is no physical reason why a particular l for an individual electron should be a valid quantum number angular momentum in classical mechanics is only conserved when there is spherical symmetry. 

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