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- Title
Novel Outlook on the Eigenvalue Problem for the Orbital Angular Momentum Operator.
- Authors
Japaridze, George; Khelashvili, Anzor; Turashvili, Koba
- Abstract
Based on the novel prescription for the power function, (x + i y) m , the new expression for Ψ (x , y | m) , the eigenfunction of the operator of the third component of the angular momentum, M ^ z , is presented. These functions are normalizable, single valued and, distinct to the traditional presentation, (x + i y) m = ρ m e i m ϕ , are invariant under the rotations at 2 π for any, not necessarily integer, m—the eigenvalue of M ^ z . For any real m the functions Ψ (x , y | m) form an orthonormal set, therefore they may serve as a quantum mechanical eigenfunction of M ^ z . The eigenfunctions and eigenvalues of the angular momentum operator squared, M ^ 2 , derived for the two different prescriptions for the square root, (m 2) 1 / 2 , (m 2) 1 / 2 = | m | and (m 2) 1 / 2 = ± m , are reported. The normalizable eigenfunctions of M ^ 2 are presented in terms of hypergeometric functions, admitting integer as well as non-integer eigenvalues. It is shown that the purely integer spectrum is not the most general solution but is just the artifact of a particular choice of the Legendre functions as the pair of linearly independent solutions of the eigenvalue problem for the M ^ 2 .
- Subjects
ANGULAR momentum (Mechanics); EIGENVALUES; LEGENDRE'S functions; NONRELATIVISTIC quantum mechanics; EIGENFUNCTIONS
- Publication
Physics (2624-8174), 2022, Vol 4, Issue 2, p647
- ISSN
2624-8174
- Publication type
Article
- DOI
10.3390/physics4020043