Classical waves

6 Assume that a particle of mass m moves only in the 

Find the force constant for the bond in each molecule. Comment on your results, since all of these molecules 

Show that a vertical spring with a mass m suspended from it is lengthened by an amount equal to mg/k 

If z is the displacement of the mass from its equilibrium position, show that the potential energy is given by 

Show that the frequency of oscillation is the same as in the horizontal position. An object of mass 0.250 kg is suspended from a spring with k = 5.55 Nm . Find the distance hy which the spring is lengthened, the period, and the frequency. 

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The miderstanding of molecular motions is necessarily based on quaiitum mechanics, the theory of microscopic physical behaviour worked out in the first quarter of the 20th century. This is because molecules are microscopic systems in which it is impossible—or at least very dangerous —to ignore the dual wave-particle nature of matter first recognized in quaiitum theory by Einstein (in the case of classical waves) and de Broglie (in the case of classical particles). 

An important ingredient in the analysis has been the positions of zeros of I (x, t) in the complex t plane for a fixed x. Within quantum mechanics the zeros have not been given much attention, but they have been studied in a mathematical context [257] and in some classical wave phenomena ([266] and references cited therein). Their relevance to our study is evident since at its zeros the phase of D(x, t) lacks definition. Euture theoretical work shall focus on a systematic description of the location of zeros. Eurther, practically oriented work will seek out computed or... 

Second, a semiclassical, or WKB, ansatz approximates the classical wave-function 4> by... 

As Bruce Clarke notes of the luminiferous/electromagnetic medium of late-classical wave theory Instead of several ethers conveying particular forms of energy, now a single ether conveyed multiple energies (Clarke 2001, 166). 

Similarities with classical waves are considered. In particular we propose that the networks of electric resonance RLC circuits may be used to study wave chaos. However, being different from quantum billiards there is a resistance from the inductors which gives rise to heat power and decoherence. 

This shows that the usual ideas associated with propagating waves in electromagnetism or fluid dynamics do not describe the behaviors found here. These differences could be expected because of the mathematical structure of reaction-diffusion equations, which owing to their parabolic character propagate information with infinite velocity. On the contrary, in the case of classical wave equations or hyperbolic equations there is a well-defined domain of influence and a characteristic velocity of propagation of information. ... 

Cross-differentiating the equations and eliminating the p yields the classic wave equation ... 

We shall apply the time-dependent perturbation theory of the last section to a system exposed to electromagnetic radiation. Before doing so, we review the classical wave theory of light.2... 

We now consider the effect of exposing a system to electromagnetic radiation. Our treatment will involve approximations beyond that of replacing (3.13) with (3.16). A proper treatment of the interaction of radiation with matter must treat both the atom and the radiation field quantum-mechanically this gives what is called quantum field theory (or quantum electrodynamics). However, the quantum theory of radiation is beyond the scope of this book. We will treat the atom quantum-mechanically, but will treat the radiation field as a classical wave, ignoring its photon aspect. Thus our treatment is semiclassical. 

First of all, consider the case when all normal vibrations are classical. This takes place if the condition a)k -4 T works well for all frequencies. In the classical case the probability of tunneling can be calculated with the help of the general formula (18) using the Franck-Condon approximation and the well-known [10] properties of quasi-classical wave functions. We will not dwell upon the details of transition from the quantum description to the... 

Looking at the phenomenon of optical absorption by the medium from the viewpoint of classical wave mechanics, we see that the attenuation of electromagnetic radiation can be attributed to the interaction of the oscillating electric vector with the medium. Any phenomenon involving periodic oscillations can be decomposed to real and imaginary components. Thus, the ordinary refractive index n is the real part of the index of refraction n, which can be written as... 

The Schrodinger equation applied the wave concept of particles to a classical wave equation yielding wavefunctions as solutions Schrodinger E (1926) Ann Phys 81 109... 

Since an electron has wave character, we can describe its motion with a wave equation, as we do in classical mechanics for the motions of a water wave or a stretched string or a drum. If the system is one-dimensional, the classical wave equation is... 

The semi-classical density expression (6.6) is adequate for computing expectations of quantities varying slowly on the scale of local wavelength. It need not be adequate when non-linear combinations of n and spatial derivatives are at issue. For this purpose, a more detailed semi-classical wave function analysis is required, or in the present context, a semi-classical analysis of (5.6) - dropping subscript v for the moment -... 

We have seen that sodium enters the cells, but potassium is set free into the extracellular space where it induces irregular depolarizations, particularly in the surroundings of an ischemic focus. Such transient depolarizations travel over the cortex like classical waves of spreading depression of electrical activity (Back et al. 1994b Leao 1944 Nedergaard and Astrup 1986). The number of such peri-infarct depolarizations (PIDs) correlated well with final... 

THE PHYSICS OF WAVES, William C. Elmore and Mark A. Heald. Unique overview of classical wave theory. Acoustics, optics, electromagnetic radiation, more. Ideal as classroom text or for self-study. Problems. 477pp. 5b 8b. 

The Helmholtz equation resembles the spatial part of the classical wave equation for matter waves (waves in ocean, sound waves, vibrations of a string, electromagnetic waves in vacuum, etc.) of amplitude F = F(r, f) ... 

The solution to this classical wave equation may be factored into space and time factors ... 

PROBLEM 3.1.1. In one dimension "derive," or give a plausibility argument, for the Schrodinger equation by combining the one-dimensional classical wave equation... 

The effect of Vq is seen to be independent of the intensity of the quantum field and to depend only on its form. This is in sharp contrast to the effect of classical waves. The effect of the quantum potential on a quantum particle has been likened to a ship on automatic pilot being guided by radio waves. Here, too, the effect of the radio waves is independent of their intensity and... 

The areas of overlap suggest constructively and destmc-tively interfering classical waves. The buildup of density in the intemuclear region of the orbital shows the connection... 

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