The q-deformed Schrödinger equation based on the q-map: one dimensional case

Abstract

In this paper, we adopt q-deformed binary operations, such as q-addition, q-subtraction, q-multiplication, and q-division, to construct the q-deformed Schrödinger equation in one dimension. We explore the mathematics involving q-deformed binary operations. We q-deform the ordinary commutator and the definition of Fock space in q-deformed quantum mechanics. As examples, we discuss the particle in a box and the harmonic oscillator.

Appendices

Appendix A

From the definition of the q-integral we have

$$\begin{aligned} \int _{[-\infty ]_q}^{[\infty ]_q } d_q x \otimes \psi ^* \otimes \psi &= \frac{1}{ q-1} \left[ q^{ \frac{q-1}{(\ln q)^2} \int _{[-\infty ]_q}^{[\infty ]_q } dx \frac{ \ln ( 1 + (q-1)\psi ^* \otimes \psi )}{ 1 + ( q-1) x} } -1 \right] \\ & = \frac{1}{ q-1} \left[ q^{ \frac{q-1}{(\ln q)^3} \int _{[-\infty ]_q}^{[\infty ]_q } dx \frac{ \ln ( 1 + (q-1)\psi ^*)\ln ( 1 + (q-1)\psi ) }{ 1 + ( q-1) x} } -1 \right] \\ &= \frac{1}{ q-1} \left[ q^{ \int _{-\infty }^{\infty } d \tilde{x} |\tilde{\psi }|^2 } -1 \right] \\ &=1. \end{aligned}$$

Appendix B

We have

$$\begin{aligned} \begin{aligned} \delta _q ({p} \ominus {p'}))&=f^{-1} \left( \delta (\tilde{p}-\tilde{p'}) \right) \\&= f^{-1} \left( \delta \left( \frac{1}{\ln q} \ln ( 1 + (q-1) p) -\frac{1}{\ln q} \ln ( 1 + (q-1) p') \right) \right) \\&= f^{-1} \left( \frac{\ln q}{q-1} ( 1 + (q-1) p) \delta ( p-p') \right) . \end{aligned} \end{aligned}$$

One can easily show the following relation,

$$\begin{aligned} \begin{aligned}&\int _{[-\infty ]_q}^{[\infty ]_q} d_q p' \otimes F(p') \otimes \delta _q ({p} \ominus {p'})= F(p)\\&\quad = \int _{[-\infty ]_q}^{[\infty ]_q} d_q p' \otimes \frac{1}{q-1} \left[ q^{ \frac{1}{(\ln q)^2} \ln ( 1 + (q-1)F(p') ) \ln ( 1 + (q-1)\delta _q ( p \ominus p') )}-1 \right] \\&\quad = \frac{1}{q-1} \left[ q^{ \frac{q-1}{(\ln q)^3} \int dp' \frac{ \ln ( 1 + (q-1)F(p') ) \ln ( 1 + (q-1) \delta _q ( p \ominus p') )}{ 1 +(q-1)p'} }-1 \right] \\&\quad = \frac{1}{q-1} \left[ q^{ \int d \tilde{p}' \tilde{F}(\tilde{p}') \delta (\tilde{p}-\tilde{p'})}-1\right] = \frac{1}{q-1} \left[ q^{ \tilde{F}(\tilde{p})}-1\right] = F(p). \end{aligned} \end{aligned}$$