Hydrogen energy eigenvalues

The energy eigenvalues of the hydrogen electronic bound states are inversely proportional to the square of the principal quantum number, in SI units,... 

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... 

This effect can be illustrated by Fig. 14.2. The effective range of local modification of the sample states is determined by the effective lateral dimension 4ff of the tip wavefunction, which also determines the lateral resolution. In analogy with the analytic result for the hydrogen molecular ion problem, the local modification makes the amplitude of the sample wavefunction increase by a factor exp( — Vi) 1.213, which is equivalent to inducing a localized state of radius r 4tf/2 superimposed on the unperturbed state of the solid surface. The local density of that state is about (4/e — 1) 0.47 times the local electron density of the original stale in the middle of the gap. This superimposed local state cannot be formed by Bloch states with the same energy eigenvalue. Because of dispersion (that is, the finite value of dEldk and... 

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

The contributions that have been considered in order to obtain precise theoretical expressions for hydrogenic energy levels are as follows the Dirac eigenvalue with reduced mass, relativistic recoil, nuclear polarization, self energy, vacuum polarization, two-photon corrections, three-photon corrections, finite nuclear size, nuclear size correction to self energy and vacuum polarization, radiative-recoil corrections, and nucleus self energy. 

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] ... 

Table 4 Energy eigenvalues for different states for the hydrogen atom confined by an impenetrable spherical box of radius tq = 2 au. The reported results were obtained by Killinbeck [32] and Friedman et al. [33] using the series method and the finite element method, respectively...

Table 7 Ground state energy eigenvalues for the hydrogen atom confined by a penetrable spherical box as a function of tq and the barrier height V0 = 0.5 hartrees. The reported results were obtained by Ley-Koo Rubinstein [30] and by Aquino [35]...

The hydrogenic orbital eigenfunctions, energy eigenvalues, and density function are determined unambiguously. 

Radial functions and energy eigenvalues for hydrogen-like atoms with this electron-nucleus potential are well-known in closed from, both in the non-relativistic and in the relativistic case. They can be found in every good textbook on quantum mechanics, for a compact reference see [48]. 

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