Bohr-Sommerfeld orbits

Lewis appropriated Bohr s new atom to try to unify the physical and chemical atom. If the Bohr-Sommerfeld orbits are in fixed positions and orientations, "they may be used as the building stones of an atom which has an essentially static character." 17 Bohr s dynamic theory works for the chemist, Lewis wrote, if the "average" position of an electron in a Bohr-Sommerfeld orbit is taken to correspond to the fixed position of the electron in Lewis s static chemical model. The outermost shell of electrons constitutes the "valence" electrons, and the remaining electrons constitute the "kernel." 18... 

Figure 2. Effect of the frequency < > of the perturbation by the core on an electron moving in a Bohr-Sommerfeld orbit of high eccentricity (low angular momentum). Plotted vs. the angle u, which varies by 2ir over one orbit. Note that the perturbation is localized near the core. In the inverse Bom-Oppenheimer limit (x 1) the perturbation oscillates many times during one orbit of the electron. (For further details and the formalism that describes the motion at high x as diffusive-like (see Refs. 3c and S.) For higher angular momentum / the effective adiabaticity parameter is x(l - e) xfl/2, where e is the eccentricity of the Bohr-Sommerfeld orbit. States of high / are thus effectively decoupled from the core.

F G. 7—4a, 6, c.—Spatial quantization of Bohr-Sommerfeld orbits withfc — 1, 2. 

This expression is nothing but the Bohr-Sommerfeld quantization rule (see, e.g., Landau and Lifshitz [1981]). In the metastable potential of Figure 3.3 there are also imaginary-time periodic orbits satisfying (3.41) that develop between the turning points inside the classically forbidden region. It is these trajectories that are responsible for tunneling [Levit et... 

The strongest pieces of evidence complex atoms provide in favour of independent electron modes and simple Bohr-Sommerfeld quantisation are (i) the existence of Rydberg series and (ii) the regularity of the periodic table of the elements. As a corollary, we should look for quantum chaos (if it occurs) in atoms for which there is some breakdown in the quality of the shell structure, combined with prolific and heavily perturbed overlapping series of interacting levels. These conditions are most readily met, as will be shown below, in the spectra of the alkaline-earth elements, as a result of d-orbital collapse. 

In the Bohr-Sommerfeld theories of the atom, the electrons are moving in orbits which are precisely specified, and the velocities are given exactly. Those theories are therefore concerned with properties which cannot be measured precisely. This difficulty is avoided, however, if one develops theories based on the wave properties of electrons we have already seen, with reference to Figure 1.1, that such theories remove some of the arbitrariness inherent in the Bohr-Sommerfeld approach. Modern theories of atoms and molecules are, therefore, wave theories, which have led to a very considerable increase in our understanding. In the remainder of this chapter we will describe aspects of wave mechanics, or quantum mechanics, that will be of help to biologists in appreciating the nature of the molecular structures with which they are concerned. 

Although the four quantum numbers n, 1, m, and s, the Pauli Exclusion Principle, and Hund s rules were developed in the context of the Bohr-Sommerfeld model, they all found immediate application to Schrodinger s new quantum mechanics. The first three numbers specified atomic orbitals (replacing Bohr s orbits). Physicist Max Bom (1882-1970) equated the square of the wave functions, to regions of probability for finding electrons in each orbital. Werner Heisenberg (1901-76), whose mathematics provide the foundation of quantum mechanics, developed the uncertainty principle the product of the uncertainty in position (Ax) of a tiny particle such as an atom (or an electron) and the uncertainty in its momentum (Ap) is larger than the quantum (h/47t) ... 

Arnold Sommerfeld (1868-1951), German physicist and professor at the Mining Academy in Clausthal, then at the Technical University of Aachen, in the key period 1906-1938, was professor at Munich University. Sommerfeld considered not only circular (Bohr-like) orbits, but also elliptical ones, and introduced the angular quantum number. He also investigated X-rays and the theory of metals. The scientific father of many Nobel Prize winners, he did not earn this distinction himself. 

Biedenharn first explains the agreement of Sommerfeld s nonrelativistic quantum numbers with the exact answer. This agreement is by no means trivial, since usually Bohr-Sommerfeld quantization rules yield quantum number which are shifted by an unknown numerical constant from the exact ones. In the nonrelativistic Kepler problem there is, however, a quantum-mechanical operator corresponding to the classical eccentricity. This makes it possible to define the spherical orbits (i.e., those with vanishing eccentricity) in an unambiguous manner, which gives an absolute frame of reference for the Bohr-Sommerfeld quantum numbers. 

Each set of these three quantum numbers (n, I, rrii) represents a valid wavefunction for the electron in a hydrogen atom. The wavefunction for a single electron in an atom is called an atomic orbital. In quantum mechanics, the position of an electron is described not in terms of orbits, as defined in the Bohr-Sommerfeld model, but in terms of its orbital. 

This clearly was, at least, a very uncomfortable situation and Sidgwick attempted to avoid the difficulty by shifting the argument away from atomic structure as such, to the idea of molecular structure in which pairs of electrons had common orbits of the Bohr-Sommerfeld type involving the molecular nuclei. He... 

In 1913 Niels Bohr proposed a system of rules that defined a specific set of discrete orbits for the electrons of an atom with a given atomic number. These rules required the electrons to exist only in these orbits, so that they did not radiate energy continuously as in classical electromagnetism. This model was extended first by Sommerfeld and then by Goudsmit and Uhlenbeck. In 1925 Heisenberg, and in 1926 Schrn dinger, proposed a matrix or wave mechanics theory that has developed into quantum mechanics, in which all of these properties are included. In this theory the state of the electron is described by a wave function from which the electron s properties can be deduced. 

I should also mention Sommerfeld, who extended Bohr s theory to try and account for the extra quantum numbers observed experimentally. Sommerfeld allowed the electrons to have an elliptic orbit rather than a circular one. 

Following Sommerfeld s proposal of elliptical electron orbits in 1915, Bohr amended his original theory, which had included only circular orbits. 14 A 1922 paper in Zeitschriftfur Physik outlined the "Aufbauprinzip" by which electrons are fed into atomic subshells. There was a neat correlation between periodic groups containing 2, 8, 8, 18,... 

An obvious possible improvement of the Bohr model was to bring it better into line with Kepler s model of the solar sxstem, which placed the planets in elliptical, rather than circular, orbits. Sommerfeld managed to solve this problem by the introduction of two extra quantum numbers in addition to the principal quantum number (n) of the Bohr model, and the formulation of general quantization rules for periodic systems, which contained the Bohr conjecture as a special case. 

The Kepler model was ceased upon by Sommerfeld to account for the quantized orbits and energies of the Bohr atomic model. By replacing the continuous range of classical action variables, restricting them to discrete values of... 

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