Position and momentum eigenfunctions - square roots of ν and δ

Kwang‐Sup Soh

David Milton Fradkin

The classical solution is used as a variable for the one-dimensional quantum-mechanical oscillator, and this leads to a separable differential equation solvable in closed form which yields a set of new quantum mechanical solutions, not energy eigenfunctions, that have the property of being localized. These new solutions are shown to be eigenfunctions of a time-independent position operator. The operator algebra of that operator and an analogous momentum operator is given, as well as the relationship of the eigenfunctions to the energy eigenfunction set. The new eigenfunctions, which are not square integrable but possess a δ-function normalization, themselves represent wavepackets that at times corresponding to when the classical position is at an extremum collapse to Dirac δ functions.

Brian P. Anderson

G. S. Japaridze, Anzor Khelashvili, K. Sh. Turashvili

Based on the novel prescription for the power function, (x+iy)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+iy)m=ρmeimϕ, are invariant under the rotations at 2π 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, (m2)1/2, (m2)1/2=|m| and (m2)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.

Won Sang Chung, M. de Montigny, HASSAN HASSANABADI

Abstract We discuss quantum mechanical systems with Dunkl derivatives by constructing the Dunkl-Heisenberg relation in the momentum representation by means of the reflection operator for momentum and we obtain the corresponding position quantum eigenfunction. We examine the one-dimensional Dunkl oscillator in the momentum space in terms of ν -deformed Hermite polynomials. We obtain the energy levels as well as the ground-state and excited wave functions in terms of the ν -deformed Hermite polynomials. We also describe some properties of the ν -deformed Hermite polynomials. We apply the method to the construction of coherent states.