Quantization, of orbitals

An effect of space quantization of orbital angular momentum may be observed if a magnetic field is introduced along what we now identify as the z axis. The orbital angular momentum vector P, of magnitude Pi, may take up only certain orientations such that the component (Pi) along the z axis is given by... 

Figure 1.9 Space quantization of orbital angular momentum for T = 3...

Apart from the assumed quantization of orbital angular momentum the Bohr model predicted the quantization of electronic energy, radius, velocity and magnetic moment of atoms ... 

Fig,3.1. Space quantization of orbital angular momenturn showing the 21+1 projections of the angular momentum vector on the z-axis. The figure is drawn for the case 1=2,... 

It can be shown quite easily that, for a filled sub-shell such as 2p or L = 0. Space quantization of the total orbital angular momentum produces 2L - - 1 components with M] = L, L —, —L, analogous to space quantization of f. In a filled sub-shell... 

To picture the spatial distribution of an electron around a nucleus, we must try to visualize a three-dimensional wave. Scientists have coined a name for these three-dimensional waves that characterize electrons they are called orbitals. The word comes from orbit, which describes the path that a planet follows when it moves about the sun. An orbit, however, consists of a specific path, typically a circle or an ellipse. In contrast, an orbital is a three-dimensional volume for example, a sphere or an hourglass. The shape of a particular orbital shows how an atomic or a molecular electron fills three-dimensional space. Just as energy is quantized, orbitals have specific shapes and orientations. We describe the details of orbitals in Section 7-1. 

However, in the sodium atom, An = 0 is also allowed. Thus the 3s —> 3p transition is allowed, although the 3s —> 4s is forbidden, since in this case A/ = 0 and is forbidden. Taken together, the Bohr model of quantized electron orbitals, the selection rules, and the relationship between wavelength and energy derived from particle-wave duality are sufficient to explain the major features of the emission spectra of all elements. For the heavier elements in the periodic table, the absorption and emission spectra can be extremely complicated - manganese and iron, for example, have about 4600 lines in the visible and UV region of the spectrum. 

The theory of color and light production, however, involves the energy levels available for electrons in atoms and molecules, according to the beliefs of modern chemical theory. In an atom or molecule, there are a number of "orbitals" or energy levels that an electron may occupy. Each of these levels corresponds to a discrete energy value, and only these energies are possible. The energy is said to be quantized, or restricted to certain values that depend on the nature of the particular atom or molecule. 

Recently, Hiipper et al. [160] have carried out a periodic-orbit quantization of the model of Ref. [159], as well as of another model [161] which predicts a shorter lifetime of about 3 fs instead of 6 fs. 

P. Gaspard To answer the question by Prof. Poliak, we expect from our present knowledge that the periodic-orbit quantization of the H + H2 dissociative dynamics on the Karplus-Porter surface can be performed with the same theory as applied to Hgl2. 

We assume that the wave functions of a set of d orbitals are each of the general form specified by 9.2-1. We shall further assume that the spin function [jj% is entirely independent of the orbital functions and shall pay no further attention to it for the present. Since the radial function R(r) involves no directional variables, it is invariant to all operations in a point group and need concern us no further. The function 0(0) depends only upon the angle 0. Therefore, if all rotations are carried out about the axis from which 0 is measured (the z axis in Fig. 8.1), (0) will also be invariant. Thus, by always choosing the axes of rotation in this way (or, in other words, always quantizing the orbitals about the axis of rotation), only the function (< ) will be altered by rotations. The explicit form of the 4>(0) function, aside from a normalizing constant, is... 

It is to be stressed that, although the two-electron submatrix elements in (14.63) and (14.65) are defined relative to non-antisymmetric wave functions, some constraints on the possible values of orbital and spin momenta of the two particles are imposed in an implicit form by second-quantization operators. Really, tensorial products (14.40) and (14.42), when the sum of ranks is odd, are zero. Thus, the appropriate terms in (14.63) and (14.65) then also vanish. 

As has been shown, second-quantized operators can be expanded in terms of triple tensors in the spaces of orbital, spin and quasispin angular momenta. The wave functions of a shell of equivalent electrons (15.46) are also classified using the quantum numbers L, S, Q, Ml, Ms, Mq of the three commuting angular momenta. Therefore, we can apply the Wigner-Eckart theorem (5.15) in all three spaces to the matrix elements of any irreducible triple tensorial operator T(JC K) defined relative to wave functions (15.46)... 

This is Coulomb s law. From the general line of argument which leads to this result, it would appear that if we had given the quantum equations (2) and (3), the spectral law (4) and if we assume so much of elementary mechanics as the energy equation (5) and the force equation for circular orbits (1) and further assume that equation (8), therefrom deduced and known to hold for an infinity of orbits, holds identically, then we are led either to a quantized force (13) or to the Coulomb law.5... 

Following the wave-mechanical reformulation of the quantum atomic model it became evident that the observed angular momentum of an s-state was not the result of orbital rotation of charge. As a result, the Bohr model was finally rejected within twenty years of publication and replaced by a whole succession of more refined atomic models. Closer examination will show however, that even the most refined contemporary model is still beset by conceptual problems. It could therefore be argued that some other hidden assumption, rather than Bohr s quantization rule, is responsible for the failure of the entire family of quantum-mechanical atomic models. Not only should the Bohr model be re-examined for some fatal flaw, but also for the valid assumptions that led on to the successful features of the quantum approach. 

The problematic part of the Bohr model is that quantization goes together with the accelerated orbital motion of an electronic particle. In order to avoid this problem an alternative explanation of orbital quantization would be required, which means discarding the particle model of the electron. 

This) space quantization of the Kepler orbits is without doubt the most surprising result of the quantum theory. The simplicity of the results and their derivation is almost like magic. 

By now, it was becoming clear that there was a connection between electrons in bodies, the radiant energy emitted by those bodies, and the distribution of that energy in the spectrum. But a more detailed theory with more information was needed. Rutherford had proposed an atom modeled on the solar system, with electrons orbiting around a positive nucleus and a lot of empty space between the electrons and the nucleus. In 1913 the Danish physicist Niels Bohr (1885-1962), who worked with Rutherford for four years and on his return to Copenhagen made Denmark a world center of theoretical physics, published one of the twentieth century s most important papers. He applied Planck s equation and the notion of quantization of energy to Rutherford s... 

---