Heisenberg-type structures of one-dimensional quantum Hamiltonians

TL;DR

This paper constructs a Heisenberg-like algebra for the one-dimensional infinite square-well potential, using non-commutative calculus, and explores its relation to generalized Heisenberg algebras including q-oscillators.

Contribution

It introduces a new algebraic structure for the infinite square-well potential using non-commutative calculus, expanding the class of generalized Heisenberg algebras.

Findings

Constructed a Heisenberg-like algebra for the square-well potential

Realized ladder operators via non-commutative differential calculus

Connected the algebra to q-oscillators and generalized Heisenberg algebras

Abstract

We construct a Heisenberg-like algebra for the one dimensional infinite square-well potential in quantum mechanics. The ladder operators are realized in terms of physical operators of the system as in the harmonic oscillator algebra. These physical operators are obtained with the help of variables used in a recently developed non commutative differential calculus. This \textquotedblleft square-well algebra\textquotedblright is an example of an algebra in a large class of generalized Heisenberg algebras recently constructed. This class of algebras also contains $q$-oscillators as a particular case. We also discuss the physical content of this large class of algebras.