Algebraic discrete quantum harmonic oscillator with dynamic resolution scaling

Abstract: We develop an algebraic formulation for the discrete quantum harmonic oscillator (DQHO) with a finite, equally-spaced energy spectrum and energy eigenfunctions defined on a discrete domain, which is known as the su(2) or Kravchuk oscillator. Unlike previous approaches, ours does not depend on the discretization of the Schr\"odinger equation and recurrence relations of special functions. This algebraic formulation is endowed with a natural su(2) algebra, each finite dimensional irreducible representation of which defines a distinct DQHO labeled by its resolution. In addition to energy ladder operators, the formulation allows for resolution ladder operators connecting all DQHOs with different resolutions. The resolution ladder operators thus enable the dynamic scaling of the resolution of finite degree-of-freedom quantum simulations. Using the algebraic DQHO formalism, we are able to rigorously derive the energy eigenstate wave functions of the QHO in a purely algebraic manner without using differential equations or differential operators, which is impossible in the continuous or infinite discrete setting. The coherent state of the DQHO is constructed, and its expected position is proven to oscillate as a classical harmonic oscillator. The DQHO coherent state recovers that of the quantum harmonic oscillator at large resolution. The algebraic formulation also predicts the existence of an inverse DQHO that has no known continuous counterpart.