Bohr-Sommerfeld model

Figure 8. A Bohr-Sommerfeld model of the xenon atom. (From H. A. Kramers and H. Horst, The Atom and the Bohr Theory of its Structure, 1924).

About one month before Heisenberg laid the formal foundation of quanmm mechanics, he wrote a letter to Bohr in which he revealed, Recently I have been occupied with the intensities [of spectral lines], notably in the case of hydrogen. The present conditions are still not entirely sufficient to obtain the intensities. The intellectual context underlying Heisenberg s remark was the Bohr-Sommerfeld model of the hydrogen atom specifically, whether the light emitted in a spectral transition is bright or dim. 

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) ... 

The selection rules given earlier for n and k were experimentally determined. The Bohr-Sommerfeld model for the atom did not account for them. Calculations based on the wave mechanical model indicate that n can change by any integer amount, as An = 0,1, 2, 3, etc., and that / must change by 1. This corresponds to the change in k that was determined experimentally. Since the wave mechanical model of the atom gives solutions in terms of statistical probability, a more accurate statement of the selection rule for / would be that the probability of a transition other than A/ = 1 is very small. 

Despite its shortcomings, however, the Bohr-Sommerfeld model is of enormous significance in the development of the quantum theory and, in a broader sense, as an illustration of the evolution of scientific ideas. 

The de Broglie relation and the Heisenberg uncertainty principle successfully danon-strated the major flaw in the Bohr-Sommerfeld model. Although Bohr went beyond classical physics in postulating the quantization of energy levels, his theory still relied... 

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. 

In this section, we intend to show that for a certain type of models the above imposed restrictions become the ordinary well-known Bohr-Sommerfeld quantization conditions [82]. For this purpose, we consider the following non-adiabatic coupling matrix x ... 

Photo 5 (left) Linus Pauling with Arnold Sommerfeld (on left). Sommerfeld, well-known professor of theoretical physics in the University of Munich, Germany, was an expert on an early form of quantum mechanics, the Bohr-Sommerfeld atomic model. The picture was taken on the occasion of Sommerfeld s visit to Caltech in 1928. Pauling studied quantum mechanics with Sommerfeld in 1926—1927, which is where Pauling got his start in the application of quantum mechanics to chemical bonding (Chapter 1) and to the calculation of molecular properties (Chapter 8). 

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... 

The models of Bohr, Sommerfeld and de Broglie provide a firm basis for the further development of a quantum theory of chemistry by re-assessment of the more advanced theories of quantum physics. However, there is little support for such a pursuit, not if we find statements like the following [14], put out by one of the world s leading publishers of academic science ... 

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... 

In Bohr s model of the hydrogen atom, the circular orbits were determined by the quantum number more accurately, by the square of the quantum number n. No other orbits were allowed. By changing the orbits from circles to ellipses, Sommerfeld introduced a second radius, which gave him another variable to play with. So it was that Sommerfeld generalized Bohr s quantum condition for electron orbits in terms of the two quantum numbers n and k. His analysis led him to establish a relationship between the two quantum numbers namely, the quantum number n set the upper limit on the quantum number k, but k could have smaller values as follows ... 

Sommerfeld s work was based on Bohr s model of the hydrogen atom. In this work, he brought relativity theory and the quantum idea together and was able to account for the fine details of the hydrogen spectrum. After Sommerfeld s paper on this work was published in 1916, he received a letter from Niels Bohr. In it Bohr wrote, I do not believe ever to have read anything with more joy than your beautiful work. ... 

Like Bohr s model of the hydrogen atom, Sommerfeld s theory flowered only briefly. The creation of quantum mechanics and the discovery of electron spin, both in 1925, followed by Paul Dirac s theory in 1928, provided a solid theory-based underpinning for... 

Arnold Sommerfeld, in 1916, set out to explain the finer details of the hydrogen spectrum. Ta do so, he refined Bohr s model by adding the possibility of elliptical orbits to the circular electron orhits Bohr had assumed in his model. This step alone, however, was not enough as the energies associated with the elliptical orbits were the same as those associated with the circular orbits. These elliptical orbits, however, did take the orbiting electron on a wild... 

It is interesting to note that straightforward Bohr-Sommerfeld quantization of the action (6.1.11) yields the exact result (6.1.25) for the bound state energies. In our units the Bohr-Sommerfeld condition results in / = n, n = 1,2,. Inserting this result into (6.1.13) indeed reproduces (6.1.25) exactly. This is the same happy accident which allowed Bohr (1913) to obtain the Balmer formula from a simple solar system model of a one-electron atom. 

Bohr s model was based on the assumption that the energy states of an atom depends on the way in which the electrons move around the nucleus. Bohr proposed that the electrons move around the nucleus in circular orbits with different radii. This model was then extended by Sommerfeld, who postulated that the orbits could be elliptical. An electron can move from one... 

Sommerfeld A (1916) The Drude dispersion theory from the standpoint of Bohr s model, and the constitution of hydrogen, oxygen, and nitrogen. Ann Phys 51 1... 

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