ABSTRACT
We discuss a finite q-model of the harmonic oscillator, based on irreducible representations of the quantum algebra https://www.w3.org/1998/Math/MathML"> s u q ( 2 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429076619/c00d8d0d-b3a0-431a-a3c2-e044ea15f7a5/content/eq1960.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . The operators of position, momentum and Hamiltonian are functions of generators of https://www.w3.org/1998/Math/MathML"> s u q ( 2 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429076619/c00d8d0d-b3a0-431a-a3c2-e044ea15f7a5/content/eq1961.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . The spectrum of position in this discrete system, in a fixed representation j of https://www.w3.org/1998/Math/MathML"> s u q ( 2 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429076619/c00d8d0d-b3a0-431a-a3c2-e044ea15f7a5/content/eq1962.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , consists of https://www.w3.org/1998/Math/MathML"> 2 j + 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429076619/c00d8d0d-b3a0-431a-a3c2-e044ea15f7a5/content/eq1963.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> 'sensor'-points https://www.w3.org/1998/Math/MathML"> x s = 1 2 [ 2 s ] q , s ∈ { - j , - j + 1 , … , j } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429076619/c00d8d0d-b3a0-431a-a3c2-e044ea15f7a5/content/eq1964.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . The momentum observable has the same spectrum. The spectrum of energies is finite and equally spaced. The wave functions are expressed in terms of dual q-Kravchuk polynomials, which are solutions to a finite-difference Schrödinger equation. Time evolution is also explicitly determined.
