Abstract
We define a new algebra, which can formally be considered as a CP deformed su(2) Lie algebra. Then, we present a one-dimensional quantum oscillator model, of which the wavefunctions of even and odd states are expressed by Krawtchouk polynomials with fixed p = 1/2, K2n(k; 1/2, 2j) and K2n(k — 1; 1/2, 2j — 2). The dynamical symmetry of the model is the newly introduced su(2)CP algebra. The model itself gives rise to a finite and discrete spectrum for all physical operators (such as position and momentum). Among the set of finite oscillator models it is unique in the sense that any specific limit reducing it to a known oscillator models does not exist.
Highlights
Krawtchouk polynomials are the simplest finite-discrete polynomials of the Askey scheme of orthogonal polynomials
Under the assumption that the quantum world is quite different from the classical one, various approaches and methods allow to construct a number of exactly solvable mathematical models of the quantum harmonic oscillator, where the wavefunctions are expressed in terms of other known orthogonal polynomials
Despite the fact that the non-relativistic one-dimensional quantum oscillator in the canonical approach has a nice explicit solution in terms of Hermite polynomials, there are still attempts to construct new oscillator models inspired by different starting points
Summary
Krawtchouk polynomials are the simplest finite-discrete polynomials of the Askey scheme of orthogonal polynomials. The Krawtchouk polynomials allow one to construct a number of interesting quantum harmonic oscillator models in finite-discrete configuration space. The current paper is structured as follows; in section 2, the CP deformation of the su(2) Lie algebra and its representations are introduced This algebra is used for the construction of a model of the finite-discrete quantum harmonic oscillator. The construction of a quantum oscillator model in term of orthogonal polynomials is based on the differential (or difference) equation, the solution of which is expressed in terms of that polynomial. Proposition 2 The symmetric Krawtchouk polynomials satisfy the following difference equations:. One observes that the spectrum of the position operator is quite different compared to the finite-discrete oscillator models already known. The main feature of the su(2)CP oscillator model is that the position values are concentrated near the border of the inteval [−j, +j], whereas near the middle of the interval the distribution is sparse
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