Draft:Hyperbolic Rosen-Morse potential

In mathematical physics, the hyperbolic Rosen–Morse potential, sometimes also called the Rosen-Morse II potential, is an exactly solvable quantum mechanical potential introduced by physicists Nathan Rosen and Philip M. Morse.

Definition and applications

Working with the dimensionless Schrödinger equation

Example plot of the hyperbolic Rosen-Morse potential

$-\psi ''(x)+V(x)\psi (x)=E\psi (x),$

a convenient parametrization of the potential, which simplifies the expressions for the solutions is

$V(x)=-\alpha (\alpha +1){\text{sech}}^{2}(x)+2\beta \tanh(x).$

Graphically, the function resembles a step potential, with a smooth potential well close to the origin. The parameters $\alpha$ and $\beta$ control the depth of the well and the height of the step, respectively. When $\beta =0$, it reduces to the Pöschl-Teller potential. When $\alpha =0$, it is equivalent to the Woods-Saxon potential.

In its full form, the hyperbolic Rosen-Morse potential appears in condensed matter physics, modeling systems with particles whose masses vary smoothly across interfaces.

Solution

The potential can be solved analytically using generalized Legendre functions. Considering the following differential equation:

$\left(1-x^{2}\right)y''-2xy'+\left[\nu (\nu +1)-{\frac {\mu ^{2}+2\mu \eta x+\eta ^{2}}{1-x^{2}}}\right]y=0,$

which reduces to Legendre's differential equation when $\eta =0$. Then, one possible solution, expressed in terms of the Gauss hypergeometric function ${}_{2}F_{1}$ is

${\displaystyle D_{\nu }^{\mu \eta }(x)=(1-x^{2})^{-{\frac {\eta }{2}}}\left({\frac {1+x}{1-x}}\right)^{\frac {\mu }{2}}{}_{2}F_{1}(-\nu -\eta ,\nu +1-\eta ;1-\mu -\eta$ ;(1-x)/2).}

If the parameters $\mu$ and $\eta$ are determined by solving the nonlinear system of equations

${\begin{cases}\mu ^{2}+\eta ^{2}=-E,\\\mu \eta =\beta ,\end{cases}}$

the solution of the Schrödinger equation is simply

$\psi (x)=D_{\alpha }^{\mu \eta }(\tanh x).$

In particular, if we have a scattering state with incoming wave number $k$:

$E=-2\beta +k^{2}.$

For $k>{\sqrt {4\beta }}$, it is possible that the wave gets transmitted through the potential. The probability of this event is the transmission coefficient $T$, given by

$T={\frac {\sinh(\pi k)\sinh(\pi {\sqrt {k^{2}-4\beta }})}{\sin ^{2}(\pi \alpha )+\sinh ^{2}\left({\frac {\pi }{2}}(k+{\sqrt {k^{2}-4\beta }})\right)}}.$

If $E<-2\beta$, normalizable bound state solutions can be found by analyzing the asymptotic behavior of the hypergeometric function. The states are indexed by an integer $n$, ranging from $0$ until the largest number satisfying

$n<\alpha -{\sqrt {\beta }}.$

The corresponding energy values are

$E_{n}=-(\alpha -n)^{2}-{\frac {\beta ^{2}}{(\alpha -n)^{2}}}.$