Quantum mechanical models in fractional dimensions

Abstract

We formulate an algebraic approach to quantum mechanics in fractional dimensions in which the momentum and position operators P, Q satisfy the R-deformed Heisenberg relations, which depend on an operator ν. We find representations of P, Q in which the dimension d and angular momentum l appear as parameters related to the eigenvalues of ν. We analyse the domain of P and find conditions which ensure that P is Hermitian. We investigate plane wave solutions and also free particle wavefunctions in fractional dimensions, and show that as a consequence of wavefunction continuity l is quantized. The representations of P, Q also lead to the corresponding representations of paraboson operators which are used to solve the harmonic oscillator in dimension d, both algebraically and analytically. We demonstrate that the formalism extends also to time-dependent Hamiltonians by solving the time-dependent harmonic oscillator in any dimension d > 0 using the method of Lewis and Riesenfeld.