Classical laws of motion

Schrodinger s equation is the wave-mechanical analogue of Hamilton s formulation of the classical laws of motion. Hamilton s function ... 

Here we derive the theorem for classical systems. The classical laws of motion can be formulated in terms of the Hamiltonian function for the particles in a system, which is defined in terms of the particle positions scalar quantities pt and qi be the entries of the vectors p and q. For a collection of N particles p e 9t3N and q e 9t3N are the collective positions and momenta vectors listing all 3/V entries. The Hamiltonian function is an expression of the total energy of a system ... 

It is possible to formulate the classical laws of motion in several ways. Newton s equations are taught in every basic course of classical mechanics. However, especially in the presence of constraint forces, the equations of motion can often be presented in a simpler form by using either Lagrangian or Hamiltonian formalism. In short, in the Newtonian approach, an /V-point particle system is described by specifying the position xa = xa(t) of each particle a as a function of time. The positions are found by solving the equations of motion,... 

We have already had before us (Chap. IV, 2, p. 69) a series of arguments to prove that the classical laws of motion cease to hold good ill the interior of atoms. We recall in particular the existence of sharp spectral lines, and the great stability of atoms, phenomena which from the classical standpoint are perfectly unintelligible. 

Thus, in Heisenberg s view, Bohr s theory fails because the fundamental ideas on which it is based (the orbit picture, the validity of the classical laws of motion, and so on) can never be put to the test. We move, therefore, in a region beyond experience, and ought not to be surprised if the theory, constructed as it is on a foundation of hypotheses which cannot be proved experimentally, partially fails in those deductions from it which can be subjected to the test of experiment. 

We have first to write down the classical laws of motion for two particles of masses m and M and charges —e and Ze which attract one another according to Coulomb s law. (These correspond exactly to the laws of motion in astronomy, except that there the attracting force is the force of gravitation.) If (Xi, z ) and x2, 2) are the... 

By knowing the position and speed of a pitched baseball and using the classical laws of motion, we can predict its trajectory and whether it will be a strike or a ball. For a baseball. Ax and Am are insignificant because its mass is enormous compared with h/Ait. Knowing the position and speed of an electron, and from them its trajectory, is another situation entirely. For example, if we take an electron s speed as 6X10 m/s 1%, then Am in Equation 7.6 is 6X10 m/s, and the uncertainty in the electron s position (Ax) is 10 m, which is about 10 times greater than the diameter of the entire atom (10 ° m) Therefore, we have no precise idea where in the atom the electron is located. 

The forces which are acting on electronic charge carriers in a liquid insulator are an external force caused by the applied electric field and an internal force originating from the interaction of the atoms or molecules comprising the liquid. The classical law of motion for an electron can then be expressed as... 

Chapter 28 offers a kinematic approach to model left ventricular diastolic function. The authors point out that kinematics-based models are species independent since they are based on classical laws of motion. They also show how such models can be validated via routine clinical methods. 

As soon as we start this journey into the atom, we encounter an extraordinary feature of our world. When scientists began to understand the composition of atoms in the early twentieth century (Section B), they expected to be able to use classical mechanics, the laws of motion proposed by Newton in the seventeenth century, to describe their structure. After all, classical mechanics had been tremendously successful for describing the motion of visible objects such as balls and planets. However, it soon became clear that classical mechanics fails when applied to electrons in atoms. New laws, which came to be known as quantum mechanics, had to be developed. 

According to the correspondence principle as stated by N. Bohr (1928), the average behavior of a well-defined wave packet should agree with the classical-mechanical laws of motion for the particle that it represents. Thus, the expectation values of dynamical variables such as position, velocity, momentum, kinetic energy, potential energy, and force as calculated in quantum mechanics should obey the same relationships that the dynamical variables obey in classical theory. This feature of wave mechanics is illustrated by the derivation of two relationships known as Ehrenfest s theorems. 

In classical mechanics the particle obeys Newton s second law of motion... 

Classical mechanics which correctly describes the behaviour of macroscopic particles like bullets or space craft is not derived from more basic principles. It derives from the three laws of motion proposed by Newton. The only justification for this model is the fact that a logical mathematical development of a mechanical system, based on these laws, is fully consistent... 

Newton s laws of motion apply to these atoms since we are treating their motion within the framework of classical mechanics. That is,... 

A. Phase Space. It will be useful here to anticipate a formulation that we will use in more detail in Section 3, namely, the solution of the classical equations of motion for the atoms of a molecule undergoing a chemical reaction. One starts with a molecule of defined geometry (say, in Cartesian coordinates) and with defined velocities for each of its atoms (expressible as components in the x, y, and z directions). The problem then is to solve Newton s second law of motion, F = mA, for each atom. The force, F, can be calculated as the first derivative of... 

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... 

In classical mechanics, Newton s laws of motion determine the path or time evolution of a particle of mass, m. In quantum mechanics what is the corresponding equation that governs the time evolution of the wave function, F(r, t) Obviously this equation cannot be obtained from classical physics. However, it can be derived using a plausibility argument that is centred on the principle of wave-particle duality. Consider first the case of a free particle travelling in one dimension on which no forces act, that is, it moves in a region of constant potential, V. Then by the conservation of energy... 

In this equation v, the vibrational quantum number, may assume the values 0, 1, 2, The frequency v. is the classical frequency of motion corresponding to this potential function it is related to the Hooke s-law constant k by the equation... 

A classical resonance-absorption theory [66, 67] was aimed to obtain the formulas applicable for calculation of the complex permittivity and absorption recorded in polar gases. In the latter theory a spurious similarity is used between, (i) an almost harmonic perturbed law of motion of a charge affected by a parabolic potential (ii) and the law of motion of a free rotor, this law being expressed in terms of the projection of a dipole moment onto the direction of an a.c. electric field. 

Much of classical physics is based on the work of Isaac Newton, who formulated three laws of motion. Because the extant mathematics of his day was not adequate to formulate these laws, he also invented a new branch of mathematics called calculus.1 Incidentally, Newton accomplished most of this work in the 18 months after he graduated from college. 

It is seen, then, that the classical picture of the electron as a compact mass in circular or elliptical motion about a heavy nucleus has little significance since such exact fixing of the position of the electron would give it a tremendous energy uncertainty. Because of the uncertainty principle, the description of the position, size, shape, and motion of the electrons in terms of everyday physics cannot be given. Rather, a new system of mechanics, quantum mechanics, yields a more satisfactory picture of certain phases of the behavior of small objects. In quantum mechanical descriptions, the ordinary laws of motion are replaced with equations representing probabilities. 

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