Quantum physics and signal processing in rigged Hilbert spaces by means of special functions, Lie algebras and Fourier and Fourier-like transforms

30th International Colloquium on Group Theoretical Methods in Physics

Celeghini, E.^1,2 ; Del Olmo, M.A.^2

Dipartmento di Fisica, Universita di Firenze, Sesto Fiorentino, Firenze

I50019, Italy^1

Departamento de Fisica Teorica, Universidad de Valladolid, Valladolid

E-47005, Spain^2

关键词: Algebraic treatment;    Discrete variables;    Heisenberg algebras;    Hermite functions;    Quantum physics;    Rigged Hilbert space;    Special functions;    Universal enveloping algebras;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/597/1/012022/pdf
DOI  :  10.1088/1742-6596/597/1/012022

来源: IOP

【 摘 要 】

Quantum physics and signal processing in the line R are strictly related to Fourier transform and Weyl-Heisenberg algebra. We discuss here the addition of a new discrete variable that measures the degree of the Hermite functions and allows to obtain the projective algebra io(2). A rigged Hilbert space is found and a new discrete basis in R obtained. All operators defined on R are shown to belong to the universal enveloping algebra of io(2) allowing, in this way, their algebraic treatment. Introducing in the half-line a Fourier-like transform, the procedure is extended to R+and can be easily generalized to Rnand to spherical cohordinate systems.