Waves wave equation

Schrodinger wave equation The fundamental equation of wave mechanics which relates energy to field. The equation which gives the most probable positions of any particle, when it is behaving in a wave form, in terms of the field. 

The miderstanding of the quantum mechanics of atoms was pioneered by Bohr, in his theory of the hydrogen atom. This combined the classical ideas on planetary motion—applicable to the atom because of the fomial similarity of tlie gravitational potential to tlie Coulomb potential between an electron and nucleus—with the quantum ideas that had recently been introduced by Planck and Einstein. This led eventually to the fomial theory of quaiitum mechanics, first discovered by Heisenberg, and most conveniently expressed by Schrodinger in the wave equation that bears his name. 

Since the potential depends only upon the scalar r, this equation, in spherical coordinates, can be separated into two equations, one depending only on r and one depending on 9 and ( ). The wave equation for the r-dependent part of the solution, R(r), is... 

The necessary boundary conditions required for E and //to satisfy Maxwell s equations give rise to tire well known wave equation for tire electromagnetic field ... 

This wave equation is tire basis of all wave optics and defines tire fimdamental stmcture of electromagnetic tlieory witli tire scalar function U representing any of tire components of tire vector functions E and H. (Note tliat equation (C2.15.5) can be easily derived by taking tire curl of equation (C2.15.1) and equation (C2.15.2) and substituting relations (C2.15.3) and (C2.15.4) into tire results.)... 

P. will of course be tire source tenn in tire wave equation. It is clear tliat for SHG tire generated polarization... 

In this seiniclassical calculation, we use only one wavepacket (the classical path limit), that is, a Gaussian wavepacket, rather than the general expansion of the total wave function. Equation (39) then takes the following form ... 

The earliest appearance of the nonrelativistic continuity equation is due to Schrodinger himself [2,319], obtained from his time-dependent wave equation. A relativistic continuity equation (appropriate to a scalar field and formulated in terms of the field amplitudes) was found by Gordon [320]. The continuity equation for an electron in the relativistic Dirac theory [134,321] has the well-known form [322] ... 

W. Greiner, Relativistic Quantum Mechanics Wave Equations, Springer-Verlag, Berlin, 1997. 

So far we have seen that a periodic function can be expanded in a discrete basis set of frequencies and a non-periodic function can be expanded in a continuous basis set of frequencies. The expansion process can be viewed as expressing a function in a different basis. These basis sets are the collections of solutions to a differential equation called the wave equation. These sets of solutions are useful because they are complete sets. 

Equation 34 has the form of the kinematic wave equation and represents a transition traveling with the wave velocity given by... 

Maxwell s equations can be combined (61) to describe the propagation of light ia free space, yielding the following scalar wave equation ... 

Any field amphtude distribution and associated propagation effects can be described equivalendy by a superposition of plane waves of appropriate amphtude and direction provided that every component plane wave satisfies equation 16. If, for example, an optical field amphtude given by the function... 

Hyperbolic The wave equation d u/dt = c d u/dx + d u/dy ) represents wave propagation of many varied types. 

In constant pattern analysis, equations are transformed into a new coordinate system that moves with the wave. Variables are changed from (, Ti) to — Ti, Ti). The new variable — Ti is equal to zero at the stoichiometric center of the wave. Equation (16-130) for a bed... 

In this chapter we define what is meant by a shock-wave equation of state, and how it is related to other types of equations of state. We also discuss the properties of shock-compressed matter on a microscopic scale, as well as discuss how shock-wave properties are measured. Shock data for standard materials are presented. The effects of phase changes are discussed, the measurements of shock temperatures, and sound velocities of shock materials are also described. We also describe the application of shock-compression data for porous media. 

Equation (4.8) is often called the shock-wave equation of state since it defines a curve in the pressure-volume plane (e.g.. Fig. 4.4). 

In order to relate the parameters of (4.5), the shock-wave equation of state, to the isentropie and isothermal finite strain equations of state (discussed in Section 4.3), it is useful to expand the shock velocity normalized by Cq into a series expansion (e.g., Ruoff, 1967 Jeanloz and Grover, 1988 Jeanloz, 1989). 

Ruoff (1967) first showed how the coefficients of the shock-wave equation of state are related to the zero pressure isentropic bulk modulus, and its first and second pressure derivatives, K q and Kq, via... 

The study of shock-wave equations of state of porous materials provides a means to expand knowledge of the equation of state of condensed materials to higher temperatures at a given volume than can be achieved along the principal Hugoniot. Materials may be prepared in porous form via pressing... 

Jeanloz, R. (1989), Shock Wave Equation of State and Finite Strain Theory, J. Geophys. Res 94, 5873-5886. 

Duvall, G.E., Shock Waves and Equations of State, in Dynamic Response of Materials to Intense Impulsive Loading (edited by Chou, P.C. and Hopkins, A.K.), US Air Force Materials Laboratory, Wright-Patterson AFB, 1973), pp. 89-121. 

From the solutions of the optical wave equations for these boundary conditions, the following statements can be verified ... 

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