Quantization, rules

In this section, we prove that the non-adiabatic matiices have to be quantized ( similar to Bohr-Sommerfeld quantization of the angulai momentum) in order to yield a continous, uniquely defined, diabatic potential matrix W(i). In another way, the extended BO approximation will be applied only to those cases that fulfill these quantization rules. The ADT matrix A(s,so) transforms a given adiabatic potential matiix u(i) to a diabatic matiix W(s, so)... 

As the D matrix is a diagonal matrix with a complex number of norm 1, the exponent of Eq. (65) has to fulfill the following quantization rule ... 

The first term in the right hand side of equation (2) dominates over 00) in the inner part of a white dwarf, while 0) becomes large in the outer part. Based on the WKB method, the quantization rule leads to... 

Equation (16) contains the information to relate the function n(r) to n(r) and it involves a factor, which changes when r equals r. There seems to be no convenient way to express this in a compact way. Its importance arises in connection with the derivation of the Bohr-Sommerfeld quantization rule. 

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... 

Tarnovskii, A.S. (1990). Bohr-Sommerfeld quantization rule and quantum mechanics, Uspekhi Fizicheskikh Nauk, 160, 155-156. [SW Phys.—Usp., 33, 86]. 

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. 

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 more general quantization rules formulated in terms of periodic action integrals of the type... 

At t = 0 the surfaces S = a, b coincide with W = a, b respectively. However, at time df the surfaces S = a, b now coincide with surfaces for which W = (a, b) + Edt. The surface S = a has therefore moved from W = a to W = a + Edt, i. e. dW = Edt. To emphasize the parallel between Sommerfeld s quantization rules and the HJ equation, the latter is reformulated [25] as a differential field equation of the action potential, W, in the same way that fluid motion is described by a velocity potential, or the propagation of a wave front. The surfaces of constant S may thus be considered as wave fronts propagating in configuration space. Let s measure the distance normal to the moving surface. Writing dW = VlT d.s, gives the velocity of the wave front... 

The Sommerfeld extension of the Bohr model was based on more general quantization rules and, although more successful at the time, is demonstrated to have introduced the red herring of tetrahedrally directed elliptic orbits, which still haunts most models of chemical bonding. 

In order to understand the occurrence of dHvA oscillations one has to take into account the quantization of the electron motion. The Bohr-Sommerfeld quantization rule for an electron in a magnetic field is... 

It is noticed that in order for the 2x2 D-matrix in Eq. (25) to become diagonal, x(s) has to fulfill the well known Bohr-Sommerfeld quantization rule of the angular momentum, namely [24]... 

In 1916. Arnold Sommerfeld generalized the Bohr quantization rule... 

The behaviour of the eigenvalues as a function of q, well documented in the literature [l4,15], is very clearly exhibited by a semiclassical analysis. Following reference [l6], it is possible to obtain by an extended WKB procedure, a quantization rule, which holds asympotically in... 

For computing the density (1.33) with ionization and affinity states (1.30) and (1.31) one uses the above second quantization rules to firstly yield the expressions... 

Clearly Sommerfeld s methods were heuristic (Bohr quantization rules), out-dated by two revolutions (Heisenberg-Schrodinger nonrelativistic quantum mechanics and Dirac s relativistic quantum mechanics) and his methods obviously had no place at all for the electron spin, let alone the four-components of the Dirac electron. So Sommerfeld s correct answer could only be a lucky accident, a sort of cosmic joke at the expense of serious minded physicists. ... 

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. 

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