Hydrogen atom energy eigenvalues

Despite the complication due to the interdependence of orbital and spin angular momenta, the Dirac equation for a central field can be separated in spherical polar coordinates [63]. The energy eigenvalues for the hydrogen atom (V(r) = e2/r, in electrostatic units), are equivalent to the relativistic terms of the old quantum theory [64]... 

This equation is the same as the angular momentum equation (5.8) of the hydrogen atom, with 2IE/H2 instead of l(l + 1). For molecular rotation it is conventional to symbolize the quantum number as J, rather than l. The energy eigenvalues are therefore determined by setting... 

In general, the Slater function is not an exact solution of any Schrodinger equation (except the Is- wavefunction, which is the exact solution for the hydrogen-atom problem). Nevertheless, asymptotically, the orbital exponent C is directly related to the energy eigenvalue of that state. Actually, at large distances from the center of the atom, the potential is zero. Schrodinger s equation for the radial function R(r) is... 

The Schrodinger operator can be used to make predictions about measurements of the energy of the electron in a hydrogen atom. For example, suppose (j) e satisfies the Schrodinger eigenvalue equation... 

En the n-th energy eigenvalue of the Schrodinger operator for the electron in the hydrogen atom, 12... 

In order to obtain the potential energy surfaces associated with chemical reactions we, typically, need the lowest eigenvalue of the electronic Hamiltonian. Unlike systems such as a harmonic oscillator and the hydrogen atom, most problems in quantum mechanics cannot be solved exactly. There are, however, approximate methods that can be used to obtain solutions to almost any degree of accuracy. One such method is the variational method. This method is based on the variational principle, which says... 

The energy eigenvalues for the confined hydrogen atom, for some states and some values of R are given in Table 1. A large range of eigenvalues is available in the literature [8,23]. The expectation values of some related dynamical quantities can be deduced [25] for these states. 

N.A. Aquino, Accurate energy eigenvalues for enclosed hydrogen atom within spherical impenetrable boxes, Int. J. Quant. Chem. 54 (2) (1995) 107-115. 

The choice of parabolic coordinates in [35] and Equation (56) is motivated by our interest in exploiting the connection between the superintegrable harmonic-oscillator and atomic-hydrogen systems [33-35]. For instance, the well-known eigenfunctions and energy eigenvalues for the two-dimensional harmonic oscillators can be written immediately by borrowing them from [33] ... 

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