The Wave Equation

In order to study the laws of wave propagation it is necessary to consider not only the disturbing force, but also the restoring force which tends to return the oscillator to its original position. The simplest kind of wave is that in which the restoring force is directly proportional to the displacement, 

In a harmonic wave the wave profile is a sine or cosine curve, thus we may write 

When equation 1.19 is substituted in formula 1.15 the expression obtained 

In a similar manner we have the solution for the cosine curve that 

The linear combination of the solutions 1.15 and 1.21 given above will also be a solution of equation 1.14. This is an example of the principle of superposition, which states that when all the relevant equations are linear we may superpose any number of individual solutions to form new functions which are themselves also solutions. The use of this principle is fundamental in the wave mechanical treatment of molecules and it will be discussed later. The linear combination here is 

From the equation of motion (Equation 2.6) and the elastic constitutive equation (Equations 2.7, 2.8), it is a simple matter to derive the wave equation, which de- 

Equating the right-hand sides of Equaticms 2.10 and 2.6 results in the wave equation for non-piezoelectric, elastic solids [3]  

Example 2.3 Derive the set of wave equations corresponding to plane wave propagation along the x direction of a cubic crystal. 

Solution The partial derivatives taken with respect to y and z are zero. Using the stiffness matrix corresponding to cubic symmetry in Equation 2.11 results in the following set of partial differential equations  

Note that the equations for , are decoupled in this case and may be solved independently. 

We shall start with an elementary and general introduction to the wave equation and become more specific and more approximate as required by the complexities to be encountered. First, we consider the Hz molecule ion because this is the simplest 

The energy of the system can be divided into potential and kinetic energy as follows  

Total energy = potential energy + kinetic energy E = P + K 

If the system were to obey classical mechanics, then E = H, where H represents the Hamiltonian for a stationary (time-independent) state. For wave motion this equation is rewritten as 

H contains both potential and kinetic energy terms. 

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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... 

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. 

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... 

A. V. Sokolov. Optical Properties ofMetaL. Elsevier, New York, 1967, Chapters 10 and 11. A very detailed, mathematical description of solutions to the wave equations, with a nice historical perspective. 

The SSW s are solutions to the wave equation in the interstitial with energy and which satisfy the following boundary conditions [3,4]. 

Wave function (Section 1.2) A solution to the wave equation for defining the behavior of an electron in an atom. The square of the wave function defines the shape of an orbital. 

The function III. 120 with more general forms of the functions u and g has also been studied in greater detail by Baber and Hasse (1937) and by Pluvinage (1950). The latter expanded g(rl2) in a power series in r12 and, by studying the formal properties of the wave equation itself, Pluvinage could derive certain general relations for the coefficients. At the Paris molecular symposium in 1957, Roothaan reported that, by expressing u and g in the form... 

In the expression (9-514), k0 = k so that in fact / ( ) is only a function ofk. In Eq. (9-514) the integration is carried out over the positive light cone. Clearly the so defined L x) will satisfy the wave equation (9-510). In order for this %u(x) to satisfy the Lorentz condition a k) must be such that... 

The wave equation representing a conservative Newtonian dynamical system is... 

The energy values correponding to the various stationary states are found from the wave equation to be those deduced originally by Bohr with the old quantum theory namely,... 

The system to be considered consists of two nuclei and one electron. For generality let the nuclear charges be ZAe and ZBe. From Born and Oppenheimer s results it is seen that the first step in the determination of the stationary states of the system is the evaluation of the electronic energy with the nuclei fixed an arbitrary distance apart. The wave equation is... 

Although no new numerical information regarding the hydrogen molecule-ion can be obtained by treating the wave equation by perturbation methods, nevertheless it is of value to do this. For perturbation methods can be applied to many systems for which the wave equation can not be accurately solved, and it is desirable to have some idea of the accuracy of the treatment. This can be gained from a comparison of the results of the perturbation method of the hydrogen molecule-ion and of Bureau s accurate numerical solution. The perturbation treatment assists, more-... 

The radicals in the denominators are necessary in order that the new eigenfunctions be normalized. The wave equation (Equation 13) can now be written... 

The wave equation for the hydrogen molecule with fixed nuclei is... 

In this case, too, molecule formation results from the symmetric eigenfunction. The corresponding perturbation energy W1 is obtained from an equation of the type of Equation 20 involving I hj and the wave equation 28. It is... 

The Interaction of Simple Atoms.—The discussion of the wave equation for the hydrogen molecule by Heitler and London,2 Sugiura,3 and Wang4 showed that two normal hydrogen atoms can interact in either of two ways, one of which gives rise to repulsion with no molecule formation, the other... 

In addition we postulate the following three rules, which are justified by the qualitative consideration of the factors influencing bond energies. An outline of the derivation of the rules from the wave equation is given below. 

The form of the functions may be closely similar to that of the molecular orbitals used in the simple theory of metals. If there are M interatomic positions in the crystal which might be occupied by any one of the N electron-pair bonds, then the M functions linear aggregates that approximate the solutions of the wave equation with inclusion of the interaction terms representing resonance. This combination can be effected with use of Bloch factors ... 

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