Rational quantum numbers

Only if shells filled sequentially, which they do not, would the theoretical relationship between the quantum numbers provide a purely deductive explanation of the periodic system. The fact the 4s orbital fills in preference to the 3d orbitals is not predicted in general for the transition metals but only rationalized on a case by case basis as I have argued. Again, I would like to stress that whether or not more elaborate calculations finally succeed in justifying the experimentally observed ground state does not fundamentally alter the overall situation.12... 

Named for the Austrian physicist Wolfgang Pauli (1900-1958), this principle can be derived from the mathematics of quantum mechanics, but it cannot be rationalized in a simple way. Nevertheless, all experimental evidence upholds the idea. When one electron in an atom has a particular set of quantum numbers, no other electron in the atom is described by that same set. There are no exceptions to the Pauli exclusion principle. 

Triplet states are, however, formed to larger extents and play more important roles than would have been thought to be true twenty or thirty years ago. The classical work of Lewis and Kasha129, whichshowed certain emissions to come from triplet states, opened the way for the rationalization of many phenomena which would otherwise prove to be quite incomprehensible. Perhaps, as Matsen etal.130 have pointed out, an undue emphasis has been placed on electron spin and on the multiplicity of states. The symmetry of the entire wave function is really the important point, and the contributions to it of all states of the molecule must be considered. Viewed in this light the triplet component , to use rather crude language, will depend on the vibrational quantum numbers in the excited state. If other isomers can exist, their contributions to the complete wave function must also be considered. 

For atoms with more than one electron, we must take account of a fourth quantum number, ms, the electron spin quantum number, which has only two values, ms = 1/2. An electron has a magnetic moment which can be rationalized by imagining that electrons spin about an... 

There were lines in the spectra corresponding to transitions other than simply between two n values (cf. Eq. 4.14). This was rationalized by Sommerfeld in 1915, by the hypothesis of elliptical rather than circular orbits, which essentially introduced a new quantum number k, a measure of the eccentricity of the elliptical orbit. Electrons could have the same n but different k s, increasing the variety of possible electronic transitions k is related to what we now call the azimuthal quantum number, Z l = k — 1). 

In Figure 2.7 the arrows indicate possible orientations of the total angular momentum vector such that the component in the line-of-force direction is always a rational fraction of the total measure. The possible vectors are identified by their projection on the radius of the unit circle as fractions k/n. The quantum number k = 0 is considered meaningless. In Sommerfeld s words [8] ... 

Visual inspection of the wavefunctions reveals another feature of the nonadiabatically delocalized states, which is crucial for rationalizing the effect and building up a model. Namely, the perpendicular quantum numbers v and V3 of the weak component of these states systematically coincide with the perpendicular quanmm numbers of an adjacent adiabatically delocalized state. This suggests that weak delocalization is induced by coupling between a certain zero-order localized state and a neighboring adiabatically delocalized state... 

These procedures are rather cumbersome, but yield a mechanism by which the dependence of potential functions of large-amplitude modes on the vibrational quantum numbers of small-amplitude modes may be rationalized. They also furnish a procedure by which the potential functions may be extrapolated to a vibrationless state if sufficient data on the vibrational dependence of the potential functions are obtained. 

With both types of vibrational spectroscopy, distinctive spectra and facility in interpretation are possible because only vibrational transitions corresponding to changes in the vibrational quantum number of+1 are allowed by the spectral selection rules. That is, An = 1, where n is the vibrational quantum number. Due to this, the frequencies observed are usually the fundamental frequencies. In addition, because of analogies between the mathematical descriptions of classical and quantum mechanical vibrating molecular systems, it is possible to rationalize many spectral observations by analogy with classical vibrating systems that possess characteristic force constants and reduced masses. This rationalization has become the basis for systematizing much of the structural and chemical information derived from vibrational spectra. 

We shall try to show that not withstanding the uncertainty which the preceding conditions contain, it yet seems possible even for atoms with several electrons to characterize their motion in a rational manner by the introduction of quantum numbers. The demand for the presence of sharp, stable, stationary states can be referred to in the language of quantum theory as a general principle of the existence and permanence of quantum numbers." ... 

Before leaving discussion of the radius, there are two other points to consider. Note that the expression for r depends on Planck s constant h. If Planck and others had not developed a quantum theory of light, the very concept of h would not exist, and Bohr would not have been able to rationalize his assumptions. A quantum theory of light was a necessary precursor to a quantum theory of matter—or at least, a theory of hydrogen. Second, the smallest value of r corresponds to a value of 1 for the quantum number n. Substituting values for all of the other constants, whose values were known in Bohr s time, one finds that for = 1 ... 

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