Quantization of the angular momentum

In this section, we prove that the non-adiabatic matrices have to be quantized (similar to Bohr-Sommerfeld quantization of the angular momentum) in order to yield a continous, uniquely defined, diabatic potential matrix W(s). 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 matrix u(s) to a diabatic matrix W(s, so)... 

Fig. 3.7. Spatial quantization of the angular momentum Ja with quantum number J" = 3 at Q-absorbtion (x = 2) of linearly polarized light with E z. The figures show the values of M" = p. The solid line indicates the spatial distribution of Ja in the classic limit (J —> oo).

This energy quantization would imply, too, quantization of the angular momentum of the electron ntevr ... 

The Bohr model of one-electron atoms Bohr postulated quantization of the angular momentum, L = m vr = nh/lir, substituted the result in the classical equations of motion, and correctly accounted for the spectrum of all one-electron atoms. E = —Z jrd (rydbergs). The model could not, however, account for the spectra of many-electron atoms. 

Two observations are to be made here. In the first place, the value of B, the Rydberg constant, appears as 2TT mEh jh and this agrees very closely indeed with that obtained by measurements on the spectral frequencies. Secondly, the characteristic form of the relation involving the factor l[n in the energy would not follow from any assumption other than one equivalent to the quantization of the angular momentum. 

The hydrogenoid orbitals have precise shapes that depend on the level of quantization of the angular momentum of the electron inside those orbitals. The angular momentum is a vector quantity. The quantization of a vector quantity is dual its intensity is quantized (takes noncontinuous absolute values), and its direction in space is quantized (takes noncontinuous orientation in space). The orbitals inside which... 

The key step occurs right here, in that Bohr solved for the velocity in terms of the velocity instead of taking the square root to find v. Thus, he used an unusual algebra step so that he could insert the quantization of the angular momentum ... 

In conclusion, we have established that the precession gives a gain of additional energy. Therefore, to treat spatial quantization of the angular momentum (7.5.3) as precession is incorrect from our point of view, because this effect does not produce any specific energy. 

An effect of space quantization of orbital angular momentum may be observed if a magnetic field is introduced along what we now identify as the z axis. The orbital angular momentum vector P, of magnitude Pi, may take up only certain orientations such that the component (Pi) along the z axis is given by... 

Just as with other angular momenta there is space quantization of rotational angular momentum so that the z component is given by... 

Quantum Number (Magnetic), A quantum number that describes the component of the angular momentum vector of an atomic electron or group of electrons in the direction of an externally applied magnetic field. The values of these components are restricted, i.e., quantized, The symbol for the magnetic quantum number is m. 

Part 2 is devoted to the foundations of the mathematical apparatus of the angular momentum and graphical methods, which, as it has turned out, are very efficient in the theory of complex atoms. Part 3 considers the non-relativistic and relativistic cases of complex electronic configurations (one and several open shells of equivalent electrons, coefficients of fractional parentage and optimization of coupling schemes). Part 4 deals with the second-quantization in a coupled tensorial form, quasispin and isospin techniques in atomic spectroscopy, leading to new very efficient versions of the Racah algebra. 

Comparing Fig. 3.6(a) and 3.6(6) we see that in both situations anisotropy with respect to M" is created in the form of alignment, since = mean value of the projection of the angular momentum upon the quantization axis equals zero. There is, however, a substantial difference. For a Q-transition, mainly the sub-levels in the vicinity of M" = 0 are populated, see Fig. 3.6(a) (bottom), whilst the level M" = 0 itself is not depopulated, as follows from (3.10). 

This expression coincides with (2.24), obtained on the basis of classical concepts. Such a coincidence is understandable from the point of view of the correspondence principle. One must bear in mind that the limit J — 00 means nothing other than that the number of projections of the angular momentum upon the z-axis, permitted by the rule of space quantization and equalling 2J + 1, becomes infinitely large, and the angular momentum becomes classical. 

Apart from the assumed quantization of orbital angular momentum the Bohr model predicted the quantization of electronic energy, radius, velocity and magnetic moment of atoms ... 

The sound part of Bohr s atomic model, and its successors, appears to be the assumed quantization of electronic angular momentum and energy, as well as atomic size. Had Bohr gone one step further the proposed quantiza-... 

A succession of levels like those of a linear molecule can be calculated for each quantum number K, which in this case describes the quantized component of the angular momentum about the unique a-axis. K cannot exceed 7, the quantum number for the total angular momentum, i.e., K = 0, 1,... dz7. For an oblate symmetric top the rotational constant A j has to be replaced by Q ]. In relation to the case of A" = 0, other K quantum numbers allowed will thus result in lower energies Ejk, which is in contrast to the prolate top with a positive term of (A[ j - 6 ]). Evidently, all rotational levels with 0 are doubly degenerate. It should be noted that each level still possesses an M-degeneracy of (27 -f 1) as discussed in connection with the linear molecule. This is due to space quantization. 

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