Quantization conditions

We concentrate on an adiabatic tri-state model in order to derive the quantization conditions to be fulfilled by the eigenvalues of the non-adiabatic coupling matrix and finally present the extended BO equation. The starting point is the 3x3 non-adiabatic coupling matrix,... 

Like the curl condition is reminiscent of the Yang-Mills field, the quantization just mentioned is reminiscent of a study by Wu and Yang [76] for the quantization of Dirac s magnetic monopole [77-78]. As will be shown, the present quantization conditions just like the Wu and Yang conditions result from a phase factor, namely, the exponential of a phase and not just from a phase. 

In Section V.A, we present a few analytical examples showing that the reshictions on the x-matrix elements are indeed quantization conditions that go back to the early days of quantum theory. Section V.B will be devoted to the general case. 

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 ... 

Next, we consider the quantization condition introduced earlier [see Eq. (94)]. Assuming F to be a circle with radius q, Eq. (94) implies... 

In the parabolic model the equations for caustics are simply Q+ = Q, and Q- = <2-- The periodic orbits inside the well are not described by (4.46), but they run along the borders of the rectangle formed by caustics. It is these trajectories that correspond to topologically irreducible contours on a two-dimensional torus [Arnold 1978] and lead to the quantization condition (4.47). 

The Bohr quantization condition, with quantum number v, corresponds to choosing the lowest Riemann sheet, which requires an additional phase correction of —2ti on crossing the branch cut, which is taken along the positive -axis. Thus the phase term cj) rises to ti at = 0+ and > 0, drops abruptly to —71 on crossing the positive fe = 0 axis, and returns to zero on the negative... 

When the Schrodinger equation for a one-electron atom is solved mathematically, the restrictions on n and I emerge as quantization conditions that correlate with energy and the shape of the wave function. 

Using these relations we can construct the bond wave functions (2) and express the matching conditions (1) in terms of the vectors (f)1,11 only. The resulting quantization conditions can be expressed in terms of the matrix,... 

These are the same quantized energy levels that arose when the wavefunction boundary conditions were matched at x = 0, x = Lx and y = 0, y = Ly. In this case, one says that the Bohr-Sommerfeld quantization condition ... 

In this section, we arrive at the quantization condition expressed in terms of periodic orbits. The periodic-orbit contribution to the trace formula can be written as the logarithmic derivative of a so-called zeta function,... 

Because of the logarithmic derivative, the poles of the resolvent appear at the zeros of the zeta functions so that we obtain the quantization condition... 

The resonances are then obtained by searching for the complex zeros of the zeta functions (4.12) in the complex surface of the energy. Assuming that the action is approximately linear, S(E,J) = T(E - Ei), while the stability eigenvalues are approximately constant near the saddle energy E, the quantization condition (4.12) gives the resonances [10]... 

In the case of symmetric molecules, we have that t - tj so that our approximate quantization condition becomes [10]... 

In Fig. 6.5 we show the eigenvalues Z, associated with several values of the quantum number nx as a function of the energy W. These values of Zx are obtained by means of the WKB quantization condition of Eq. (6.26). For a constant value... 

If we are only interested in the frequency of the modulations in the vicinity of the zero field limit we may employ a different approach, used by Freeman et al,6,7 and Rau10. They used the fact that the motion in the direction is bound and found the energy separation between successive eigenvalues. Specifically, they used Eq. (8.8), the WKB quantization condition for the bound motion in the direction, and differentiated it to find the energy spacing between states of adjacent n1 or, equivalently, between the oscillations observed in the cross sections. Differentiating Eq. (8.8) with respect to energy yields... 

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... 

Applying the semiclassical quantization condition to the action around a closed classical orbit yields25-27... 

Bohr s theory was extended in various ways, especially by Somerfeld, who showed how to deal with elliptical orbits. There was a certain amount of qualitative success in applying the theory to atoms with several electrons. These developments in what is now called the old quantum theory were important as they laid much of the groundwork necessary for a correct theory. Ultimately, they were unsuccessful. Bohr s theory did not really explain what is going on why should only some orbits be allowed Where does the quantization condition (eqn 4.12) come from Following the developments of... 

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