Angular momentum magnitude

The rules for allowed angular momentum magnitudes and orientations are the same for the rigid rotor as for the hydrogenlike ion. This leads to the following relationships in terms of the rotational quantum numbers J = 0,1,2,..., mj =... 

Now, we have besides the vibrational, the electronic angular momentum the latter is characterized by the quantum number A corresponding to the magnitude of its projection along the molecular axis, L. Here we shall consider A as a unsigned quantity, that is, for each A 7 0 state there will be two possible projections of the electronic angular momentum, one corresponding to A and the other to —A. The operator Lj can be written in the form... 

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... 

From a quantum mechanical treatment the magnitude of the angular momentum due to the spin of one electron, whether it is in the hydrogen atom or any other atom, is given by... 

Similarly, the magnitude of the angular momentum Pj due to nuclear spin is given by... 

Each electron in an atom has two possible kinds of angular momenta, one due to its orbital motion and the other to its spin motion. The magnitude of the orbital angular momentum vector for a single electron is given, as in Equation (1.44), by... 

For an electron having orbital and spin angular momentum there is a quantum number j associated with the total (orbital + spin) angular momentum which is a vector quantity whose magnitude is given by... 

The vector L is so strongly coupled to the electrostatic field and the consequent frequency of precession about the intemuclear axis is so high that the magnitude of L is not defined in other words L is not a good quantum number. Only the component H of the orbital angular momentum along the intemuclear axis is defined, where the quantum number A can take the values... 

The nuclei of many isotopes possess an angular momentum, called spin, whose magnitude is described by the spin quantum number / (also called the nuclear spin). This quantity, which is characteristic of the nucleus, may have integral or halfvalues thus / = 0, 5, 1, f,. . . The isotopes C and 0 both have / = 0 hence, they have no magnetic properties. H, C, F, and P are important nuclei having / = 5, whereas and N have / = 1. 

The last particular case worthy of analysis is an anticorrelated process. The Keilson-Storer parameter y, when negative, describes relaxation induced by collisions, which primarily change the direction of the angular momentum to its opposite. In other words, at y = — 1 the only result of a collision is that the direction of free rotation becomes reversed without changing the magnitude of the angular velocity. Substituting... 

Consider now spin-allowed transitions. The parity and angular momentum selection rules forbid pure d d transitions. Once again the rule is absolute. It is our description of the wavefunctions that is at fault. Suppose we enquire about a d-d transition in a tetrahedral complex. It might be supposed that the parity rule is inoperative here, since the tetrahedron has no centre of inversion to which the d orbitals and the light operator can be symmetry classified. But, this is not at all true for two reasons, one being empirical (which is more of an observation than a reason) and one theoretical. The empirical reason is that if the parity rule were irrelevant, the intensities of d-d bands in tetrahedral molecules could be fully allowed and as strong as those we observe in dyes, for example. In fact, the d-d bands in tetrahedral species are perhaps two or three orders of magnitude weaker than many fully allowed transitions. 

Finally we observe from Fig. 1 the magnitude of Gw qz) decreases for increasing at fixed w. Thus, the only way to fulfill the Bethe sum rule at arbitrarily large values of q will be to include basis functions of arbitrarily high angular momentum. This confirms a previously reached conclusion [12],... 

Since the linear velocity vector v, is perpendicular to the radius vector r the magnitude Z, of the angular momentum is... 

The area A enclosed by the circular electronic orbit of radius r is rcr. From equation (5.63) we have the relation L = mecor. Thus, the magnitude of the magnetic moment is related to the magnitude L of the angular momentum by... 

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