Half theory fractional angular momentum and the application of fractional derivatives to quantum mechanics

The periodic table provides a deep unifying principle for understanding chemical behaviour by relating the properties of different elements. For those belonging to the fifth and earlier rows, the observations concerning these properties and their interrelationships acquired a sound theoretical basis by the understanding of electronic behaviour provided by non-relativistic quantum mechanics. However, for elements of high nuclear charge, such as occur in the sixth and higher rows of the periodic table, the systematic behaviour explained by non-relativistic quantum mechanics begins to fail. These problems are resolved by realizing that relativistic quantum mechanics is required in heavy elements where electrons velocities can reach significant fractions of the velocity of light. An essentially non-mathematical description of relativistic quantum mechanics explains how relativity modifies valence electron behaviour in heavy elements. The direct relativistic effect, arising from the relativistic increase of the electron mass with velocity, contracts orbitals of low angular momentum, increasing their binding energies. The indirect relativistic effect causes valence orbitals of high angular momentum to be more effectively screened as a result of the relativistic contraction of the core orbitals. In the alkali and alkaline earths, the s orbital contractions reverse the chemical trends on descending these groups, with heavy elements becoming less reactive. For valence d and f electrons, the indirect relativistic effect enhances the reductions in their binding energies on descending the periodic table. The d electrons in the heavier coinage metals thus become more chemically active, which causes these elements to exhibit higher oxidation states. The indirect effect on d orbitals causes the chemistries of the sixth-row transition elements to differ significantly from the very similar behaviours of the fourth and fifth-row transition series. The relativistic destabilization of f orbitals causes lanthanides to be chemically similar, forming mainly ionic compounds in oxidation state three, while allowing the earlier actinides to show a richer range of chemical behaviour with several higher oxidation states. For the 7p series of elements, relativity divides the non-relativistic p shell of three degenerate orbitals into one of much lower energy with the energies of the remaining two being substantially increased. These orbitals have angular shapes and spin distributions so different from those of the non-relativistic ones that the ability of the 7p elements to form covalent bonds is greatly inhibited. This article is part of the theme issue ‘Mendeleev and the periodic table’.

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Fractional angular momentum in noncommutative generalized Chern-Simons quantum mechanics

The European Physical Journal Plus ◽

10.1140/epjp/i2016-16229-9 ◽

2016 ◽

Vol 131 (7) ◽

Cited By ~ 1

Author(s):

Xi-Lun Zhang ◽

Yong-Li Sun ◽

Qing Wang ◽

Zheng-Wen Long ◽

Jian Jing

Keyword(s):

Quantum Mechanics ◽

Angular Momentum ◽

Chern Simons

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The Quantum Mechanics of Cohesion

Collection of Czechoslovak Chemical Communications ◽

10.1135/cccc20070252 ◽

2007 ◽

Vol 72 (2) ◽

pp. 252-268

Author(s):

Roy McWeeny

Keyword(s):

Wave Function ◽

Quantum Mechanics ◽

Angular Momentum ◽

Scalar Product ◽

Hydrogen Molecule ◽

Fundamental Problem ◽

Ionic Species ◽

General Context ◽

Spin Angular Momentum ◽

Actual Nature

We consider the fundamental problem of "what makes atoms stick together in molecules, crystals, or clusters?" The Heitler and London paper (1927) on the hydrogen molecule marked a first attempt to discuss, in terms of quantum mechanics, the interaction of two atoms with unpaired spins. The aim of this note is to show how the primitive concepts used eighty years ago still retain a certain validity even in a much more general context. We consider in fact the interaction of two arbitrary systems, each with a resultant spin angular momentum, and show how the interaction energy depends on the scalar product of the two resultants. The actual nature of the two systems is irrelevant: they may be atoms, molecules, or ionic species of any kind each described by a wave function which may be, in principle, exact. This provides a first step in the formulation of any general theory of cohesion.

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