Bohr-Sommerfeld quantization condition

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 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 ... 

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... 

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... 

Extensive information on weakly bound, physisorbed atoms on solids is available from bound state resonance data obtained from beam scattering experiments. Strong changes in the scattering intensities are observed when selective adsorption in the physisorption well occurs (see, for instance, ref. [113]). The resonance positions yield the eigenvalues E of the potential V(z). These satiesfies to a good approximation the Bohr-Sommerfeld quantization condition... 

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 applied a quantum condition to the energy states of the hydrogen atom only certain energy states were allowed. Sommerfeld applied a quantum condition to the orientation of electron orbits only certain spatial orientations relative to an applied magnetic field were allowed. The experiment of Stem and Ger-lach was designed to test Sommerfeld s explanation of the Zeeman effect, namely, the idea of space quantization. 

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