Bohr-Sommerfeld quantization

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)... 

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 ... 

These are the same quantized energy levels that arose when the wavefunetion boundary eonditions were matehed at x = 0, x = Lx and y = 0, y = Ly. In this ease, one says that the Bohr-Sommerfeld quantization eondition ... 

By contrast, in heavy-light-heavy molecules such as HMuH, C1HC1, or IHI, a very extended elliptic island exists in the classical phase space [150]. In such cases, the elliptic island may be the support of several metastable states that can be obtained by Bohr-Sommerfeld quantization. Their lifetime is determined by dynamical tunneling from inside the elliptic island to the outside regions. 

As pointed out by Edmonds and Starace,12,13 the atoms are excited near the origin and can only escape in the z directions. The motion in the x,y plane is bound and is most likely to be the source of the quasi Landau resonances. To find the locations of the resonances it is adequate to ignore the z motion entirely and simply compute the energy spectrum of the motion in x,y plane. Applying the Bohr-Sommerfeld quantization condition 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]. 

Before 1925/26 quantum mechanics was based on ad hoc quantization procedures such as the Bohr-Sommerfeld quantization method. The disadvantage of this method is that it is coordinate dependent, and usually works only in Cartesian coordinates. In 1917 Einstein suggested a coordinate independent quantization procedure, an important improvement over the Bohr-Sommerfeld quantization scheme. Einstein s quantization method was subsequently extended and improved by Brillouin (1926) and Keller (1958). Therefore, this quantization scheme is referred to as EBK... 

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. 

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]... 

Semiclassical SCF makes the following additional simplifications.10 The semiclassical Bohr-Sommerfeld quantization is applied to obtain the singlemode energies ... 

For a diatom (as for a separable vibrational mode in a polyatomic) the product vibrational quantum number is found from the Bohr-Sommerfeld quantization conditions namely that pr dr = (v 4- 1/2)h for bound motions (27). That is, if the momentum is followed over one half-period the product vibrational action can be calculated ... 

The spectral density function A r, r e) is clearly related to the wave functions in the WKB approximation. In particular, in case (iii), it is the product of two connecting solutions across a turning point. The Green s function does not, at this level of approximation, give the Bohr-Sommerfeld quantization condition, but one can note that in case (ii) the spectral density function is the product of the wave fimctions in the end points of the interval and an amplitude factor, which assumes its extremum value when the quantization condition... 

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