The classical wave equation

At the other end of the segment the tensile force acts in the opposite direction, and we have 

The net perpendicular force on our string segment is the resultant of these two  

The difference in slope at two infinitesimally separated points, divided by dx, is by definition the second derivative of a function. Therefore, 

Equation (1-13) gives the force on our string segment. If the string has mass m per unit length, then the segment has mass m dx, and Newton s equation F=ma may be written 

Equation (1-14) is the wave equation for motion in a string of uniform density under tension T. It should be evident that its derivation involves nothing fundamental beyond Newton s second law and the fact that the two ends of the segment are linked to each other and to a common tensile force. Generalizing this equation to waves in three-dimensional media gives 

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

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 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) ... 

In this chapter, the most important quantum-mechanical methods that can be applied to geological materials are described briefly. The approach used follows that of modern quantum-chemistry textbooks rather than being a historical account of the development of quantum theory and the derivation of the Schrodinger equation from the classical wave equation. The latter approach may serve as a better introduction to the field for those readers with a more limited theoretical background and has recently been well presented in a chapter by McMillan and Hess (1988), which such readers are advised to study initially. Computational aspects of quantum chemistry are also well treated by Hinchliffe (1988). 

Although the Schrddinger equation, the fundamental equation of quantum mechanics, may be derived from the classical wave equation by a heuristic approach, it is currently more common to present quantum mechanics as a set of postulates, such as those tabulated below (Levine, 1983). 

II. The Classical Wave Equation for Radiation Fields in Nonlinear Media... 

II. THE CLASSICAL WAVE EQUATION FOR RADIATION FIELDS IN NONLINEAR MEDIA... 

THE CLASSICAL WAVE EQUATION The classical law governing wave motion is the wave equation... 

Finally, a remark or two about the treatment at the end of Chapter 19, which attempts to relate the classical wave equation and the Schrodinger equation. It should be clear that whether or not the Schrodinger equation is correct depends only on its predictions of behavior and not in the least on whether or not there is some means of transforming the classical wave equation into the Schrodinger equation. On the other hand, the Schrodinger treatment of a system is required to reduce to Newtonian mechanics in the limit as Planck s constant approaches zero, or in the limit of large masses and distances. Suffice it to say that the Schrodinger equation does reduce properly in these circumstances. 

In order to sustain the note and to create music, only those wavelengths that have an amplitude of zero at both ends of the fixed string can constructively interfere with one another to create what is known as a standing wave. A standing wave (also called a stationary wave) Is a wave that exists in a fixed position. Although the standing wave appears to be stationary, it is actually oscillating up and down in place as a function of time. Therefore, the amplitude of the wave (y) is a function of its position (x) and time (t), as shown in Equation (3.3). This is the classical wave equation in one-dimension (see Appendix A for a derivation). 

Quantum mechanics is a model that is based entirely on postulates that explain the observations associated with atomic and subatomic particles. As such, the Schrodinger equation cannot be derived from first principles. However, what follows is a rationale for the Schrodinger equation. The classical wave equation in one-dimension was given by Equation (3.3) and is reproduced in Equation (3.34) with the substitution of T for y. 

Hence, the classical wave equation in one-dimension becomes ... 

Principle of circularity. The principle of circularity, applied to the loop of paths in this Formal Graph, immediately provides a general wave equation, which has therefore the same shape whatever the energy variety. (The classical wave equation is found at the condition of being within the restricted framework of the linearity of system properties.) The link between the property of space time and the system properties stems from this circularity. In this systan, this link is created by the relationship between the propagation velocity (velocity of phase) and the ratio of the mechanical tension on the lineic mass. 

Setting the right-hand side to zero, equation (1.4) turns into the classical wave equation, the prototype linear differential equation wifli harmonic waves as solutions. The full equation (1.4) is known to have solitary solutions (x,r) = < >(x — vf), where v is the velocity of the solitary wave and... 

In classical mechanics we have separate equations for wave motion and particle motion, whereas in quantum mechanics, in which the distinction between particles and waves is not clear-cut, we have a single equation— the Schrodinger equation. We have seen that the link between the Schrddinger equation and the classical wave equation is the de Broglie relation. Let us now compare Schrodinger s equation with the classical equation for particle motion. 

The wavefunctions for time-independent states are eigenfunctions of Schrodinger s equation, which can be constructed from the classical wave equation by requiring X = h/ /2m E — V), or from the classical particle equation by replacing pk W i v h/2ni)d/dk, k = x,y, z. 

The classical wave equation describes the vibrations of strings and light waves. 

In the formal theory of quantum mechanics, the Schrodinger wave equation is taken as a postulate (fundamental hypothesis). In order to demonstrate a relationship with the classical wave equation, we obtain the time-independent Schrodinger equation nonrigorously for the case of a particle that moves parallel to the x axis. For a standing wave along the x axis, the classical coordinate wave equation of Eq. (14.3-10) is... 

There is no way to obtain the time-dependent Schrodinger equation from a classical wave equation. The classical wave equation of a vibrating string, Eq. (14.3-3), is second order in time. It requires two initial conditions (an initial position and an initial velocity) to make a general solution apply to a specific case. The uncertainty principle of quantum mechanics (to be discussed later) implies that positions and velocities cannot be specified simultaneously with arbitrary accuracy. For this reason only one initial condition is possible, which requires the Schrodinger equation to be first order in time. The fact that the equation is first order in time also requires that the imaginary unit i must occur in the equation in order for oscillatory solutions to exist. 

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