A hydrogen-like atom

A quantum analogue of the Sun-Earth system is a nucleus and one electron, i.e. a hydrogen-like atom. The force holding the nucleus and electron together is the Coulomb interaction. 

The interaction again only depends on the distance, but owing to the small mass of the electron, Newton s equation must be replaced with the Schrodinger equation. For bound states, the time-dependence can be separated out, as shown in Section 1.6.1, giving the time-independent Schrddinger equation. 

The Hamiltonian operator for a hydrogen-like atom (nuclear charge of Z) can in Cartesian coordinates and atomic units be written as eq. (1.33), with M being the nuclear and m the electron mass (m = 1 in atomic units). 

The two kinetic energy operators are already separated, since each only depends on three coordinates. The potential energy operator, however, involves all six coordinates. The centre of mass system is introduced by the following six coordinates. 

Here the X, Y, Z coordinates define the centre of mass system, and the x, y, z coordinates specify the relative position of the two particles. In these coordinates the Hamiltonian operator can be rewritten as eq. (1.36). 

---

It is customary to express the empirical data concerning term values in the X-ray region by introducing an effective nuclear charge Zeff e in the place of the true nuclear charge Ze in an equation theoretically applicable only to a hydrogen-like atom. Often a screening constant S is used, defined by the equation... 

Stark effect of a hydrogen-like atom, using the Schrodinger wave mechanics. Their equation, obtained independently and by different methods, is... 

According to the discussion in Section II, the quantity TT" represents the electron density about the nucleus in a hydrogen-like atom. The electron... 

A transition metal with the configuration t/ is an example of a hydrogen-like atom in that we consider the behaviour of a single (d) electron outside of any closed shells. This electron possesses kinetic energy and is attracted to the shielded nucleus. The appropriate energy operator (Hamiltonian) for this is shown in Eq. (3.4). 

Figure 6.7 A typical series of spectral lines for a hydrogen-like atom shown in terms of the wave number v.

When a magnetic field B is applied to a hydrogen-like atom with magnetic moment M, the resulting potential energy V is given by the classical expression... 

The postulates 1 to 6 of quantum meehanies as stated in Sections 3.7 and 7.2 apply to multi-particle systems provided that each of the particles is distinguishable from the others. For example, the nucleus and the electron in a hydrogen-like atom are readily distinguishable by their differing masses and charges. When a system contains two or more identical particles, however, postulates 1 to 6 are not sufficient to predict the properties of the system. These postulates must be augmented by an additional postulate. This chapter introduces this new postulate and discusses its consequences. 

The first term on the right-hand side is just the energy of a hydrogen-like atom with nuclear charge Z, namely, —Z e jiao. The third term has the same value as the first. The second term is evaluated as follows... 

Show that in a hydrogen-like atom of nuclear charge Ze the average distance of the electron from the nucleus, in the stale described by quantum numbers /, >i, is... 

The distinction in standard non-relativistic theory between spin-orbit interaction as relativistic on the one hand and other spin interactions as non-relativistic on the other hand does lead to some inconsistencies. Consider, for instance, a hydrogen-like atom where the coordinate system is shifted from the... 

Let the atoms in the chain be numbered 0, 1,. . . , iV, and let the foreign atom be denoted by X (Fig. 1). Associated with each atom we introduce an atomic orbital < (r, m). These orbitals are divided into two sets. One set (m = X) contains only one member, which is the orbital on the foreign atom the other set (m = 0, 1,. . . , A) consists of the orbitals on the crystal atoms. Thus, we have the problem of the interaction of a hydrogen-like atom with a crystal whose normal electronic structure consists of just one band of states. 

As an example we apply the variational principle to the evaluation of the ground state energy of a hydrogen-like atom using a minimum basis set of two-component radial functions ... 

As the wave function is not known analytically for systems larger than a hydrogen-like atom, suitable approximate wave functions have to be found and the accuracy of Eq. (1) depends of course on the level of approximation. A survey of the various quantum chemical methods to generate approximated wave functions can be found in Refs. (22,23). Here, we shall only present the foundations of Hartree-Fock and density functional theory (DFT) needed in later sections. 

The same argument can be made for each alkali atom because there is only one outer electron, one can model an alkah atom as a hydrogen-like atom with one electron and a nucleus made up of the true nucleus and the inner electrons. As above, this argument hinges on the fact that the inner electrons tend to be in the lowest possible states, while the Pauli exclusion principle forbids any two electrons from occupying the same state. And indeed, spectral data for alkali atoms resembles spectral data for hydrogen. Moreover, the chemical properties of the alkali atom is similar. For example, each combines easily with chlorine to form a salt such as potassium chloride, lithium chloride... 

When the wave equation for a hydrogen-like atom is solved in the most direct way for orbitals with the angular momentum quantum number / = 3, the following results are obtained for the purely angular parts (i.e., omitting all numerical factors) ... 

For a hydrogen-like atom or ion the spin-orbit coupling constant i is expressed as... 

This is the most stable orbital of a hydrogen-like atom—that is, the orbital with the lowest energy. Since a Is orbital has no angular dependency, the probability density 2 is spherically symmetrical. Furthermore, this is true for all s orbitals. We depict the boundary surface for an electron in an s orbital as a sphere (Figure 1-2). The radial function ensures that the probability for finding the particle goes to zero for r — °°. 

Fig. 5. Diagrams contributing to the AC Stark-shifts and mixing of the I1S1/2- and 2S i/2-levels of a hydrogen-like atom under the action of two laser waves of the frequency u>l, counter-propagating in the direction nz. As seen by an atom in its rest frame moving with the velocity v = vn relative to the laboratory system, appropriate Doppler-shifted frequencies are determined as 0)1,2 = wl(1 vn-n /c) / 1 — (v/c)2...

The g Factor of a Bound Electron in a Hydrogen-Like Atom... 

Abstract. Recently, a precise measurement on the bound electron g factor in hydrogenlike carbon was performed [1]. We consider the present status of the theory of the g factor of an electron bound in a hydrogen-like atom and discuss new opportunities and possible applications of the recent experiment. 

When this expression is extended to many-electron systems, two related problems arise. Firstly, what is the effective spin-orbit hamiltonian for the electron in open shells Secondly, what is the potential in which they move For a hydrogen-like atom the field would be written... 

A well known operator is the Hamiltonian of an electron in centre-of-mass coordinates of a hydrogen-like atom... 

---