Angular Momentum Calculus In Quantum Physics

Author : Steven Bentley Bildstein
Publisher :
ISBN 13 :
Total Pages : 123 pages
Book Rating : 4.:/5 (112 download)

DOWNLOAD NOW!

Download or read book Fractional Angular Momentum and the Application of the Fractional Calculus to Quantum Mechanics written by Steven Bentley Bildstein and published by . This book was released on 2019 with total page 123 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this dissertation, we develop the fractional derivative and apply it to various problems in Quantum Mechanics. Among other things, we find fractional angular momentum, with “Fractional Spherical Harmonics” as solutions to the squared Quantum Mechanical angular momentum operator, but with non-integer eigenvalues. These fractional functions might be interpreted as intermediate states visited by an orbiting electron as it jumps between the more stable, integer angular momentum values in an atom. For example, we will find that the azimuthal part of the fractional spherical harmonic takes a form which precesses and experiences a radial acceleration, unlike the single-valued, standard solution . One might conclude that this acceleration causes the fractional state to be transient. We will find that the eigenvalues of these fractional spherical harmonics form a spectrum which is a hybrid between being discrete and continuous. The spectrum forms orthogonal towers in the quantum number l, indexed by a given parameter. Upon selecting the continuous real number parameter, the fractional spherical harmonics, for this parameter, are orthogonal in the discrete index l. Furthermore, inserting these angular eigenstates into Hamiltonians with an angular momentum term, we find the resulting eigenfunctions of the Hamiltonian to also have a hybrid orthogonal structure, with the standard discrete eigenvalues indexed by the continuous parameter. In this dissertation, we show that this generated parameter can be put to use, through an extremization process, to improve the accuracy in variational and perturbation calculations. We use fractional hydrogenic wave functions as trial functions in a variational calculation of the least upper bound to the ground states of the di-Hydrogen ion and the Helium atom. We find an improvement over the corresponding technique by about 4% for the ion, and approximately 0.02% for the Helium atom. When the “best” value for parameter, obtained by minimizing the above energy estimate, is substituted into the fractional eigenfunction, an integrable singularity, or “spike”, in the polar direction is created. We have called this 1 spike a “probe”, which might influence atomic orientation in the formation of a given physical lattice. In the attempt to apply these fractional techniques to perturbation theory, we find even better improvement. We outline this technique in the dissertation and apply it to two perturbed systems which can be solved exactly. This enables a direct comparison between our fractional, and the standard method. One of these perturbed systems is the finestructure of Hydrogen where the ratios of the error, dE(frac), of the fractional first-order calculation, and the error, dE(std), of the corresponding standard first-order calculation vary between approximately, 0.175