Wave equation, form

The most useful sets of orthogonal functions for our purposes are the wave functions belonging to a given wave equation. In preceding chapters we have shown that the solutions of certain wave equations form sets of normalized orthogonal functions, such as for example the Hermite orthogonal functions which... 

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 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] ... 

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

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... 

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... 

Strittmater (S6) has presented a solution to the damped acoustic wave equation, and shown that the acoustic pressure has the form... 

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 ... 

Schrodinger (IV) has tentatively advanced a form of the wave equation in which magnetic fields are considered. 

In the VB method, a wave equation is written for each of various possible electronic structures that a molecule may have (each of these is called a canonical form), and the total )/ is obtained by summation of as many of these as seem plausible, each with its weighting factor ... 

This resembles Eq. (1.1), but here each j/ represents a wave equation for an imaginary canonical form and each c is the amount contributed to the total picture by that form. For example, a wave function can be written for each of the following canonical forms of the hydrogen molecule " ... 

The two chief general methods of approximately solving the wave equation, discussed in Chapter 1, are also used for compounds containing delocalized bonds. In the VB method, several possible Lewis structures (called canonical forms) are drawn and the molecule is taken to be a weighted average of them. Each in Eq. (1.3),... 

This was first found by Schrodinger in 1926 starting with Eq. (2), which he called the eigentliche Wellengleichung. (Paradoxically, this got translated to real wave equation [2].) In the form... 

In the derivation used here, it is clear that two approximations have been made—the configurations are incoherent, and the nuclear functions remain localized. Without these approximations, the wave function form Eq. (C.l) could be an exact solution of the Schrodinger equation, as it is in 2D MCTDH form (in fact is in what is termed a natural orbital form as only diagonal configurations are included [20]). 

We know that the general form of the wave equation is... 

In order to preserve the resemblance to Schrodinger s equation Dirac obtained another relativistic wave equation by starting from the form... 

For wave functions like = exp[if x,t)], the squared operator would mask the phase information, since = <3> 2, and to avoid this, a linear Schrodinger operator would be preferred. This has the immediate advantage of a wave equation which is linear in both space and time derivatives. The most general equation with the required form is... 

Multiplet structures are ignored completely and electronic configurations are defined only in terms of the occupation numbers of the various orbitals. Accordingly, the radial HFS wave equations for a free atom or ion are written in the form... 

Because of the interelectronic repulsion term l/ri2, the electronic Hamiltonian is not separable and only approximate solution of the wave equation can be considered. The obvious strategy would be to use Hj wave functions in a variation analysis. Unfortunately, these are not known in functional form and are available only as tables. A successful parameterization, first proposed by James and Coolidge [89] and still the most successful procedure, consists of expressing the Hamiltonian operator in terms of the four elliptical coordinates 1j2 and 771 >2 of the two electrons and the variable p = 2ri2/rab. The elliptical coordinates 4> 1 and 2, as in the case of Hj, do not enter into the ground-state wave function. The starting wave function for the lowest state was therefore taken in the power-series form... 

Time-Independent Wave Packet Forms of Schrodinger and Lippmann-Schwinger Equations. 

In a line of reasoning that many of the younger quantum physicists regarded as reactionary, Schrodinger built his treatment of the electron on the well-understood mathematical techniques of wave equations as partial differential equations involving second derivatives. Schrodinger s equation for stationary electron states, as written in the Annalen der Physik in 1926, took the form... 

In papers , unsteady-state regime arising upon propagation of the stationary fundamental mode from linear to nonlinear section of a single-mode step-index waveguide was studied via numerical modeling. It was shown that the stationary solution to the paraxial nonlinear wave equation (2.9) at some distance from the end of a nonlinear waveguide has the form of a transversely stable distribution ( nonlinear mode ) dependent on the field intensity, with a width smaller than that of the initial linear distribution. 

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