Hydrogenlike atoms

Normalized radial functions for a hydrogenlike atom are given in Table A 1.1 and plotted graphically in Fig. A 1.1 for the first ten combinations of n and /. It will be seen that the radial functions for Is, 2p, 3d, and 4f orbitals have no nodes and are everywhere of... 

Energy levels of a hydrogenlike atom of atomic number Z ... 

The expectation values of various powers of the radial variable r for a hydrogenlike atom with quantum numbers n and I are given by equation (6.69)... 

The normalized radial wavefunctions for hydrogenlike atoms can be expressed by... 

This formula for hydrogenlike atoms with nuclear charge Z, is often formulated in terms of the dimensionless fine-structure constant a = 4Tre0hc/e2 as... 

The largest Lyapunov exponents for the relativistic hydrogenlike atom in a uniform magnetic field... 

Theoretically, light muonic atoms have two main special features as compared with the ordinary electronic hydrogenlike atoms, both of which are connected with the fact that the muon is about 200 times heavier than the electron First, the role of the radiative corrections generated by the closed electron loops is greatly enhanced, and second, the leading proton size contribution becomes the second largest individual contribution to the energy shifts after the polarization correction. 

The wave functions for a state of a hydrogenlike atom described by the quantum numbers n (total quantum number), l (azimuthal quantum number), and m (magnetic quantum number) are usually expressed in terms of the polar coordinates r, 8, and . The orbital wave function is a product of three functions, each depending on one of the coordinates ... 

Figure 6-3 Schematic diagram of the relative energies of the hydrogenlike atomic orbitals...

FIGURE 1.25 The radial wave-functions of the first three s-orbitals of a hydrogenlike atom. Note that the number of radial nodes increases (as n — l), as does the average distance of the electron from the nucleus. Because the probability density is given by ijr2, all s-orbitals correspond to a nonzero probability density at the nucleus. 

Another type of interactions that can support an infinite number of bound states has an attractive dipole potential h2a/(2[zR2) asymptotically. Here, a is some constant, R is the distance between the centers of mass of the two subsystems which the whole system separates into, and /x is their reduced mass. Such an interaction occurs between a charged particle and a hydrogenlike atom in an excited state [59, 60]. 

Equation 4.13 expresses the total (kinetic plus potential) energy of the electron of a hydrogenlike atom in terms of four fundamental quantities of our universe electron charge, electron mass, the permittivity of empty space, and Planck s constant. From Eq. 4.13 the energy change involved in emission or absorption of light by a hydrogenlike atom is simply... 

The Bohr approach works well for hydrogenlike atoms, atoms with one electron hydrogen, singly-ionized helium, doubly-ionized lithium, etc. However, it showed... 

The standard approach to solving the Schrodinger equation for hydrogenlike atoms involves transforming it from Cartesian (x, y, z) to polar coordinates (r, 6, (p), since these accord more naturally with the spherical symmetry of the system. This makes it possible to separate the equation into three simpler equations, fir) = 0, fid) = 0, and fiip) = 0. Solution of the fir) equation gives rise to the n quantum number, solution of the/(0) equation to the l quantum number, and solution of the fifi) equation to the mm (often simply called m) quantum number. For each specific n = n, 1 = 1 and mm = inm there is a mathematical function obtained by combining the appropriate fir), fi fJ) and /([Pg.101]

The calculation yields v = 2.19 x 10s ms The value of v is correct for hydrogenlike atoms (one electron), because for these the Bohr atom is a correct model, at least mathematically if not conceptually. It should be approximately right for atoms with more than one electron, because we are considering n = 1, an s electron, and the effect of outer-shell electrons on the first shell is not large. This velocity is 2.19 x 108/3.00 x 10s = 0.73 of the speed of fight. 

In the non-relativistic quantum mechanics the nuclear recoil effect for a hydrogenlike atom is easily taken into account by using the reduced mass p = mM/(m + M) instead of the electron mass m (M is the nuclear mass). It means that to account for the nuclear recoil effect to first order in m/M we must simply replace the binding energy E by E(1 — m/M). 

Let us consider now a relativistic hydrogenlike atom. In the infinite nucleus mass approximation a hydrogenlike atom is described by the Dirac equation (h = c = l)... 

First attempts to derive formulas for the relativistic recoil corrections to all orders in aZ were undertaken in [11,12]. As a result of these attempts, only a part of the desired expressions was found in [12] (see Ref. [13] for details). The complete aZ-dependence formula for the relativistic recoil effect in the case of a hydrogenlike atom was derived in [14]. The derivation of [14] was based on using a quasipotential equation in which the heavy particle is put on the mass shell [15,16]. According to [14], the relativistic recoil correction to the energy of a state a is the sum of a lower-order term ALL and a higher-order term A Eh ... 

Table 1. The results of the numerical calculation of the function P aZ) for low-lying states of hydrogenlike atoms...

Table 1. The loop-after-loop contribution to the second-order Lamb shift of the ground state of hydrogenlike atoms expressed in terms of the function Gui(Za) defined by Eq. (2)...

In a particular hydrogenlike atom, spectral lines appear with wavelengths 19.440 0.007 nm and 17.358 0.005 nm. What is the maximum probable error in the difference between these wavelengths ... 

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