Scattering of a scalar relativistic particle by the hyperbolic tangent potential

Abstract

We solve the Klein-Gordon equation in the presence of the hyperbolic tangent potential. The scattering solutions are derived in terms of hypergeometric functions. The reflection, R, and transmission, T, coefficients are calculated in terms of gamma function and, superradiance is discussed, when the reflection coefficient, R, is greater than one.Résumé : Nous solutionnons l'équation de Klein-Gordon en présence d'un potentiel tangent hyperbolique. Les solutions de diffusion sont écrites à l'aide de fonctions hypergéométriques. Nous calculons les coefficients de réflexion R et de transmission T en termes de fonctions gamma et nous discutons de super-radiance, lorsque le coefficient R est plus grand que un.

Klein paradox for bosons, wave packets and negative tunnelling times

“…for which the transmission amplitude was given in [20]. Now the potential varies over a region δx ∼ 1/b, and the Klein tunnelling persists for as long as δx is small compared to the particle's Compton wavelength, δx < /mc.…”

“…Now the potential varies over a region δx ∼ 1/b, and the Klein tunnelling persists for as long as δx is small compared to the particle's Compton wavelength, δx < /mc. Using the results of [20] it can be shown (see Methods) that for a smooth potential (19), with δx << d, one can continue using the MRE (13), with the pole in T n (p, −q) moved into the complex p-plane,…”

“…we use the connection rules for a smooth step W(x) = V tanh(bx) in 20 , namely where where Ŵ(z) is the Gamma function 21 , q and p are the momenta at x → ±∞ , and…”

Study of superradiance in the Lambert-W potential barrier

“…The scattering of a spin-0 particle by a one-dimensional potential barrier is a typical problem that appears in relativistic quantum mechanics and has been the subject of much interest in recent years. [1][2][3][4][5][6][7][8][9][10] Consider an incoming wave from left to right; the common situation is that the wave loses energy because of its interaction with the potential barrier; therefore, the incoming amplitude is greater than the amplitude of the reflected wave. 11 The amplitude of the incident wave, T , is called the transmission coefficient, and the amplitude of the reflected wave, R, is called the reflection coefficient.…”

“…The superradiance phenomenon has been widely discussed in the literature for the Dirac equation [12][13][14][15][16][17] and for the Klein-Gordon equation. [8][9][10] This phenomenon also appears in astrophysics in the scattering of scalar waves by rotating black holes. 11,18 In this article, we study the phenomenon of superradiance, when the reflection coefficient R is greater than one, for the Lambert-W potential barrier.…”