Hydrogen-like atom energy levels

The hydrogen-like atomic energy levels are given in equation (6.48). If n and 2 are the principal quantum numbers of the energy levels E and E2, respectively, then the wave number of the spectral line is... 

Solving equation (6.21) for the energy E and replacing A by n, we obtain the quantized energy levels for the hydrogen-like atom... 

The energy levels of the hydrogen-like atom depend only on the principal quantum number n and are given by equation (6.48), with replaeed by ao, as... 

The theoretical results for the hydrogen-like atom may be related to experimentally measured spectra. Observed spectral lines arise from transitions of the atom from one electronic energy level to another. The frequency v of any given spectral line is given by the Planck relation... 

The situation is similar to when one used to deal with an object of a simple structure of the states, but a complecated nature. There are a number of examples of such objects in atomic, nuclear and particle physics. An effective hamiltonian, described with a few parameters, is usually introduced and the parameters must be determined experimentally. A well-known example is the nuclear contributions to the atomic energy levels in the hydrogen-like atoms... 

The term Lamb shift of a single atomic level usually refers to the difference between the Dirac energy for point-like nuclei and its observable value shifted by nuclear and QED effects. Nuclear effects include energy shifts due to static nuclear properties such as the size and shape of the nuclear charge density distribution and due to nuclear dynamics, i.e. recoil correction and nuclear polarization. To a zeroth approximation, the energy levels of a hydrogen-like atom are determined by the Dirac equation. For point-like nuclei the eigenvalues of the Dirac equation can be found analytically. In the case of extended nuclei, this equation can be solved either numerically or by means of successive analytical approximation (see Rose 1961 Shabaev 1993). 

Table 7 Perturbation theory coefficients E for expansion of the energies in powers of R for the Is, 2p, 3d, 4f and 5g levels of a hydrogen-like atom confined in a spherical box of radius R (see Equation (51) of text). The numbers in parentheses indicate the power of 10... 

An important generalization of the quantum theory by Sommerfeld [125] and independently by Wilson [142] allowed a detailed study of the non-radiating non-circular orbits, and led to Sommerfeld s celebrated fine structure formula which represents the energy levels of hydrogen-like atoms to a precision which was substantiated by the most refined experiments over the twenty years following its derivation. Comparison with experiment, however, implies a consideration, not only of energy levels, but also of the relative intensities of spectral lines. We shall see that on this point the theory failed. 

Rydberg constant - The fundamental constant which appears in the equation for the energy levels of hydrogen-like atoms i.e., = hcR where h is Planck s constant, c... 

Rydberg constant (R - The fundamental constant which appears in the equation for the energy levels of hydrogen-like atoms i.e., E = hcR. 2 JrP-, where h is Planck s constant, c the speed of light, Z the atomic number, (Xthe reduced mass of nucleus and electron, and n the principal quantum number (n = 1,2,. ..). 

The direct variational method has been used to solve Schrodinger equation (4.2) with respect to Eqs. (4.3), (4.4), and (4.5). Hydrogen-like atom wave functions 2p and 3p have been taken as trial functions for ground and exited states respectively. Two-fermion wave functions were written in conventional form as a product of coordinate part and symmetric or asymmetric spin part for triplet (spin S = 1) or singlet (spin E = 0) state. The energy level positions 23 (S = 0), 33 (S = 0),... 

The energy levels of the hydrogen-like atom are degenerate because more than one state corresponds to a specific value of n (the principal quantum number). These degenerate states are characterized by the quantum numbers I and m, which characterize the spherical harmonics of the wave function. For each value of n there are n values of l(l = 0,.,., n — 1) and for each value of / there are 2/ -h 1 values of m(m = — / -h 1,..., 0,..., / — 1, /) giving... 

Atoms and molecules can adsorb and emit radiation to change their internal energy states. The electronic transitions of the hydrogen-like atoms have already been mentioned. The quantization of the energy levels restricts the possible wavelengths of the radiation to discrete spectral lines. Only certain transitions are allowed and these are given by separate selection rules for electronic. 

The Coulomb potential, such as that for the hydrogen atom, resembles vaguely the Morse curve. Yet its form is a little similar to the Morse potential (dissociation limit, but infinite depth). We expect, therefore, that the energy levels for the hydrogen-like atom will become closer and closer when the energy increases, and we are right. Is the number of these energy levels finite as for the Morse potential This is a more subtle question. Whether the number is finite or not is decided by the asymptotics (the behaviour at infinity). The Coulomb potential makes the number infinite. 

FIGURE 12.4 The effect of more than one electron on the electronic energy levels of an atom. For hydrogen-like atoms, all of the energy levels with the same principal quantum number n are degenerate. For atoms having more than one electron, the shells are separated by the quantum number as well. (Energy axis is not to scale.)... 

However, this is misleading. Although electronic energy levels are dictated by the principal quantum number, we should remember that a principal quantum shell in a hydrogen atom has other quantum numbers, namely, f and m. If the symmetries of the operator and wavefunctions in equation 15.1 were examined, one would find that it is the angular momentum quantum number that dictates the selection rule. The specific selection rule for allowed electronic transitions in the hydrogen atom (or, for that matter, hydrogen-like atoms) is... 

---