The $\mathfrak{su}(2)$ Krawtchouk oscillator model under the ${\cal C}{\cal P}$ deformed symmetry

TL;DR

This paper introduces a new ${ m CP}$-deformed $rak{su}(2)$ algebra and constructs a finite quantum oscillator model with wavefunctions expressed via Krawtchouk polynomials, featuring unique spectral properties.

Contribution

It defines a novel ${ m CP}$-deformed $rak{su}(2)$ algebra and develops a finite oscillator model with unique spectral characteristics and wavefunctions based on Krawtchouk polynomials.

Findings

The model has a finite, discrete spectrum for physical operators.

Wavefunctions are expressed by Krawtchouk polynomials with fixed parameters.

The model is unique among finite oscillator models, with no known limiting cases.

Abstract

We define a new algebra, which can formally be considered as a ${\cal C}{\cal P}$ deformed $\mathfrak{su}(2)$ 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 $p=1/2$, $K_{2n}(k;1/2,2j)$ and $K_{2n}(k-1;1/2,2j-2)$. The dynamical symmetry of the model is the newly introduced $\mathfrak{su}(2)_{{\cal C}{\cal P}}$ 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.