Angular quantization

In addition to affecting the number of active degrees of freedom, the fixed n also affects the iinimolecular tln-eshold E in). Since the total angular momentum j is a constant of motion and quantized according to... 

Thus, I and m quantize the vibrational angular momentum and its z component. 

The relationship of these quantum meehanieal operators to experimental measurement will be made elear later in this ehapter. For now, suffiee it to say that these operators define equations whose solutions determine the values of the eorresponding physieal property that ean be observed when a measurement is earried out only the values so determined ean be observed. This should suggest the origins of quantum meehanies predietion that some measurements will produee discrete or quantized values of eertain variables (e.g., energy, angular momentum, ete.). 

An identity that proves very useful when treating eoupled angular momenta that are subjeeted to rotations of the axes with respeet to whieh their eigenfunetions are quantized ean be derived by eombining the above result ... 

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... 

Figure 1.9 Space quantization of orbital angular momentum for T = 3...

Figure 1.10 Space quantization of electron spin angular momentum...

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

As we saw in Equation (1.61), space quantization of rolational angular momenlum of a diatomic or linear polyatomic molecule is expressed by... 

Equation (1.48) shows that, for I =, space quantization of nuclear spin angular momentum results in the quantum number Mj taking the values 5 or — 5. The nuclear spin wave function J/ is usually written as a or /i, corresponding to Mj equal to 5 or —5,... 

It can be shown quite easily that, for a filled sub-shell such as 2p or L = 0. Space quantization of the total orbital angular momentum produces 2L - - 1 components with M] = L, L —, —L, analogous to space quantization of f. In a filled sub-shell... 

At low energies, the rotational and vibrational motions of molecules can be considered separately. The simplest model for rotational energy levels is the rigid dumbbell with quantized angular momentum. It has a series of rotational levels having energy... 

Eigenstates of a crystal, 725 Eigenvalues of quantum mechanical angular momentum, 396 Electrical filter response, 180 Electrical oscillatory circuit, 380 Electric charge operator, total, 542 Electrodynamics, quantum (see Quantum electrodynamics) Electromagnetic field, quantization of, 486, 560... 

The quasi-classical theory of spectral shape is justified for sufficiently high pressures, when the rotational structure is not resolved. For isotropic Raman spectra the corresponding criterion is given by inequality (3.2). At lower pressures the well-resolved rotational components are related to the quantum number j of quantized angular momentum. At very low pressure each of the components may be considered separately and its broadening is qualitatively the same as of any other isolated line in molecular or atomic spectroscopy. 

The constant h and the hypothesis that energy is quantized in integral multiples of hv had previously been introduced by M. Planck (1900) in his study of blackbody radiation. In terms of the angular frequency a> deflned in equation (1.2), the energy E of a photon is... 

In the previous section the g value was considered as a scalar quantity, which was indeed a good approximation since the unpaired electron on the hydrogen atom occupies a spherically symmetric s orbital. If the unpaired electron exhibits p or d character the electron possesses both spin and orbital angular momentum. As a result the spin is not quantized exactly along the direction of the external field and the g value becomes a tensor... 

A complete decomposition of the ab initio computed CF matrix in irreducible tensor operators (ITOs) and in extended Stevens operators. The parameters of the multiplet-specific CF acting on the ground atomic multiplet of lanthanides, and the decomposition of the CASSCF/RASSI wave functions into functions with definite projections of the total angular momentum on the quantization axis are provided. 

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)... 

This is precisely the relationship that was required when Bohr assumed that the angular momentum of the electron is quantized for the allowed orbits. 

The first application of quantum theory to a problem in chemistry was to account for the emission spectrum of hydrogen and at the same time explain the stability of the nuclear atom, which seemed to require accelerated electrons in orbital motion. This planetary model is rendered unstable by continuous radiation of energy. The Bohr postulate that electronic angular momentum should be quantized in order to stabilize unique orbits solved both problems in principle. The Bohr condition requires that... 

The principle of NMR can be explained in quantum mechanical terms. The angular momentum of a spinning nucleus, quantized both in magnitude and direction, is given by the equation... 

In this section, we shall look at the way these various absorptions are analysed by spectroscopists. There are four kinds of quantized energy translational, rotational, vibrational and electronic, so we anticipate four corresponding kinds of spectroscopy. When a photon is absorbed or generated, we must conserve the total angular momentum in the overall process. So we must start by looking at some of the rules that allow for intense UV-visible bands (caused by electronic motion), then look at infrared spectroscopy (which follows vibrational motion) and finally microwave spectroscopy (which looks at rotation). 

---