Scattering of electrons and positrons from argon and krypton in the GUP framework

Abstract

We study the scattering of electrons and positrons by argon and krypton atoms considering a generalized uncertainty principle (GUP). To proceed, we first prove the modified non-relativistic wave equation due to the minimal length in the GUP framework and then determine the wave function of incident particles applying a scattering potential including the static and the energy-dependent correlation–polarization contributions. Finally, we compute the transmission and reflection probabilities from potential barrier. It will be shown that some exotic phenomena occur in this framework which cannot be justified by the ordinary quantum mechanics.

Appendix

In this appendix, the wave function $\phi (x)$ satisfying the boundary conditions (4) and (5) is constructed from the particular amplitude-phase solutions discussed in Sect. 2. For the left side of the barrier, the solution $\phi_{L}^{ - } (x)$ can be written as

$$\lim_{x \to - \infty } \phi_{L}^{ - } \approx \Omega_{L} (x)e^{{i( - S_{L} x + \delta_{L} )}}$$

(26)

where $\delta_{{\text{L}}} = - \lim_{x \to - \infty } (U_{{\text{L}}} - S_{{\text{L}}} x)$ and $\Omega_{{\text{L}}} ( - \infty ) = 1/\sqrt {S_{{\text{L}}} }$.

Using the matrix (8), one also has

$$\lim_{x \to + \infty } \phi_{R}^{ \pm } (x) \approx \Omega_{R} (x){\text{e}}^{{ \pm i(S_{R} x + \delta_{R} )}}$$

(27)

where $\delta_{R} = \lim_{x \to \infty } (U_{R} - S_{R} x)$ and $\Omega_{R} (\infty ) = 1/\sqrt {S_{R} }$.

According to the matrix (8) and the relations (26) and (27), one obtains

$$\begin{aligned} \phi _{{\text{L}}}^{ - } (x) \approx & \frac{{\omega _{2} - i\omega _{0} }}{{2\sqrt {S_{{\text{R}}} } }}\text{e} ^{{ + i(S_{{\text{R}}} x + \delta _{{\text{R}}} )}} \\ & + \frac{{\omega _{1} + i\omega _{0} }}{{2\sqrt {S_{{\text{R}}} } }}\text{e} ^{{ - i(S_{{\text{R}}} x + \delta _{{\text{R}}} )}} ,\quad x \to + \infty \\ \end{aligned}$$

(28)

Normalizing (28) according to Eq. (5), for the transmission and reflection amplitudes, one finds the following formulae

$$\begin{gathered} t = 2\left( {i\omega_{0} + \omega_{1} } \right)^{ - 1} {\text{Exp}}\left[ {i\left( {\delta_{{\text{L}}} + \delta_{{\text{L}}} } \right)} \right] \hfill \\ r = \left( {\omega_{2} - i\omega_{0} } \right)\left( {i\omega_{0} + \omega_{1} } \right)^{ - 1} {\text{Exp}}[2i\delta_{{\text{R}}} ]. \hfill \\ \end{gathered}$$

(29)

where the phases $\delta_{{\text{R}}}$ and $\delta_{{\text{L}}}$ are real. According to the definition of the T-coefficient and R-coefficient, one has

$$\begin{gathered} T = |t|^{2} = 4\left( {\omega_{0}^{2} + \omega_{1}^{2} } \right)^{ - 1} \hfill \\ R = |r|^{2} = \left( {\omega_{0}^{2} + \omega_{2}^{2} } \right)\left( {\omega_{0}^{2} + \omega_{1}^{2} } \right)^{ - 1} . \hfill \\ \end{gathered}$$

(30)