The Wave Mechanics

Schrodinger s equation is the wave-mechanical analogue of Hamilton s formulation of the classical laws of motion. Hamilton s function  

Although wave equations are readily composed for more-electron atoms, they are impossible to solve in closed form. Approximate solutions for many-electron atoms are all based on the assumption that the same set of hydrogen-atom quantum numbers regulates their electronic configurations, subject to the effects of interelectronic repulsions. The wave functions are likewise assumed to be hydrogen-like, but modified by the increased nuclear charge. The method of solution is known as the self-consistent-field procedure. 

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Calculate the value of the first three energy levels according to the wave mechanical picture of a particle in a one-dimensional box. Take the case of nitrogen... 

Morse P M 1929 Diatomic molecules according to the wave mechanics II. Vibrational levels Phys. Rev. 34 57... 

Schutte, C. J. H. (1968) The Wave Mechanics of Atoms, Molecules and Ions, Arnold, London. 

The Wave Mechanics of the Surface Bond in Chemisorption T. B. Grimley... 

This assumption is no longer valid in its place the wave mechanics provides the simple explanation that the repulsive forces arise from the interpenetration of the atoms. As a simple example, we may consider the hydrogen ion and the chloride ion according to the wave mechanics the potential energy of these two ions at a distance R apart, assuming that no deformation occurs, is6... 

Such an action is understandable with the wave-mechanics picture of interpenetrating ions. 

In the wave mechanics of Schrodingerf a conservative Newtonian dynamical system is represented by a wave function or amplitude function obtained from the partial differential equation... 

Added February 10, 1927.—J. H. Van Vleck in Proc. Nat. Acad. America, vol. 12, p. 662 (December, 1926), has discussed the mole refraction and the diamagnetic susceptibility of hydrogen-like atoms with the use of the wave mechanics, obtaining results identical with our equations (24) and (34). He also considered the effect of the relativity corrections (which is equivalent to the effect of a central field) and concluded that equation (24), derived by the use of parabolic instead of spherical co-ordinates, is not invalidated.]... 

Hartree, DR. The wave mechanics of an atom with a non-coulomb central field. 

The wave mechanics discussed in Chapter 2 is a linear theory. In order to develop the theory in a more formal manner, we need to discuss the properties of linear operators. An operator 4 is a mathematical entity that transforms a function ip into another function 0... 

To represent observables in n-dimensional space it was concluded before that Hermitian matrices were required to ensure real eigenvalues, and orthogonal eigenvectors associated with distinct eigenvalues. The first condition is essential since only real quantities are physically measurable and the second to provide the convenience of working in a cartesian space. The same arguments dictate the use of Hermitian operators in the wave-mechanical space of infinite dimensions, which constitutes a Sturm-Liouville problem in the interval [a, 6], with differential operator C(x) and eigenvalues A,... 

In summary, the wave mechanical treatment of the hydrogen atom assumes that the electronic motion is described by tp, obtained as solutions of the equation (in atomic units)... 

For systems with classical analogues every observable quantity is an algebraic function of <7, and p . The wave-mechanical operator acting on the -function is... 

The filled (Is)2 cores lead to additional steric repulsions with the incoming donor hybrid. These combine with nuclear Coulomb repulsions to oppose high electronic overlap. Figure 3.6 displays the calculated increase in steric repulsion as each filled 2s spin-orbital of Li2 collides with the filled core Is of the opposite atom. The paired one-electron steric repulsions shown in Fig. 3.6 are similar to the ionic two-electron steric repulsions of Fig. 2.11, and the wave-mechanical origin of the steric pressure would be analogous to that described in the discussion surrounding Fig. 2.12. 

The Occurrence of Her mite Functions in Wave Mechanics. The Hcrmite functions which wc have discussed in the Inst section occur in the wave mechanical treatment of the harmonic oscillator1). Although this is a very simple mechanical system the analysis of its properties is of great importance because of its application to the quantum theory of radiation. 

Morse, P. M. (1929), Diatomic Molecules According to the Wave Mechanics. II. Vibrational Levels, Phys. Rev. 34, 57. 

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