The Schrodinger Hydrogen Atom

In this chapter we hope that the previous rigid rotor derivation will pay off. Here we only have to solve the radial equation since we already know the (0)di (c[)) = TJ (9, 4 ) solutions. We now have to relax the rigid potential to permit the Coulomb attraction between the electron and the nucleus. 

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In the absence of the second proton, Eq. (7.19) is the Schrodinger equation for the free hydrogen atom. The ground-state wavefunction is ... 

One-body systems of particular interest are the free spinless particle and the spinless particle bound in a Coulomb potential Uc (the nonrelativistic hydrogen atom). Spin is introduced in section 3.3.2. The state of a free spinless particle is an eigenstate of momentum. It is completely specified by the momentum p. The corresponding Schrodinger equation is... 

REVIEW OF THE SUPERINTEGRABILITY OF THE SCHRODINGER EQUATION FOR THE FREE HYDROGEN ATOM AND ITS IONIZATION LIMIT... 

The ionization limit of the Schrodinger equation and its eigenfunctions for the free hydrogen atom, at a vanishing energy value, corresponds to Bessel functions in the radial coordinate as known in the literature and illustrated in 2.1. The counterparts for paraboloidal [21], hyperboloidal [9], and polar angle [22] coordinates have also been shown to involve Bessel functions. These limits and their counterparts for the other coordinates are reviewed successively in this section. 

Although it is typically solved in spherical and parabolic coordinates, the hydrogenic Schrodinger equation is also separable in prolate spheroidal coordinates, with one focus at the nucleus and the other located along the Lenz vector at a distance R away. These coordinates are ordinarily used for two- center problems such as H. Previously, general features of the spheroidal hydrogen atom have been explored by Coulson and Robinson [6], who noted that the limits R —> 0 and R —> oo yield the spherical and parabolic solutions, respectively. Demkov [7] used the spheroidal eigenfunctions to construct... 

Table 6.4 Energy eigenvalues for the Dirac and Schrodinger hydrogen atom in Hartree atomic units. The Dirac eigenvalue , and the shifted energy , were calculated for c = 137.037. Then nteC takes a value of 18779.139369 in Hartree atomic units for this value of the speed of light. The difference between , and " is rather small and can only be enhanced for larger values of the nuclear charge number Z. The difference is the larger the smaller n and j are.

Is the fundamental relation of atomic spectroscopy inherently able to confirm the validity of the assumptions and thus that of the derived energy equations of the nonrelativistic hydrogen-atom models of Bohr, Schrodinger, and Heisenberg ... 

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. 

I 1 11 Schrodinger equation can be solved exactly for only a few problems, such as the particle in a box, the harmonic oscillator, the particle on a ring, the particle on a sphere and the hydrogen atom, all of which are dealt with in introductory textbooks. A common feature of these problems is that it is necessary to impose certain requirements (often called boundary... 

The Hydrogenic atom problem forms the basis of much of our thinking about atomic structure. To solve the corresponding Schrodinger equation requires separation of the r, 0, and (j) variables... 

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