Lambert-W step-potential

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The Lambert-W step-potential^[1] affords the fifth – next to those of the harmonic oscillator plus centrifugal, the Coulomb plus inverse square, the Morse, and the inverse square root^[2] potentials – exact solution to the stationary one-dimensional Schrödinger equation in terms of the confluent hypergeometric functions.^[3] The potential is given as

V(x)={\frac {V_{0}}{1+W(e^{-x/\sigma })}}

where $W$ is the Lambert function also known as the product logarithm. This is an implicitly elementary function that resolves the equation $We^{W}=z$ .

The Lambert $W$ -potential is an asymmetric step of height $V_{0}$ whose steepness and asymmetry are controlled by parameter $\sigma$ . If the space origin and the energy origin are also included, it presents a four-parametric specification of a more general five-parametric potential which is also solvable in terms of the confluent hypergeometric functions. This generalized potential, however, is a conditionally integrable one (that is, it involves a fixed parameter).

Solution

The general solution of the one-dimensional Schrödinger equation for a particle of mass $m$ and energy $E$ :

{\frac {d^{2}\psi }{dx^{2}}}+{\frac {2m}{\hbar ^{2}}}(E-V(x))\psi =0

for the Lambert $W$ -barrier for arbitrary $V_{0}$ and $\sigma$ is written as

\psi (x)=z^{i\delta /2}e^{-isz/2}\left({\frac {du(z)}{dz}}-i{\frac {\delta +s}{2}}u(z)\right),z=W(e^{-x/\sigma })

where $u(z)$ is the general solution of the scaled confluent hypergeometric equation

u''(z)+\left({\frac {i\delta }{z}}-is\right)u'(z)+{\frac {as}{z}}u(z)=0

and the involved parameters are given as

a={\frac {\delta (\delta +s)}{2s}}+{\frac {\sigma {\sqrt {m}}V_{0}}{{\sqrt {2E}}\hbar }},\delta =2\sigma {\sqrt {\frac {2m(E-V_{0})}{\hbar ^{2}}}},s=2\sigma {\sqrt {\frac {2mE}{\hbar ^{2}}}}

A peculiarity of the solution is that each of the two fundamental solutions composing the general solution involves a combination of two confluent hypergeometric functions.

If the quantum transmission above the Lambert $W$ -potential is discussed, it is convenient to choose the general solution of the scaled confluent hypergeometric equation as

u=c_{1}(isz)^{1-i\delta }{}_{1}F_{1}(1+i(a-\delta );2-i\delta ;isz)+c_{2}U(ia;i\delta ;isz)

where $c_{1,2}$ are arbitrary constants and ${}_{1}F_{1}$ and $U$ are the Kummer and Tricomi confluent hypergeometric functions, respectively. The two confluent hypergeometric functions are here chosen such that each of them stands for a separate wave moving in a certain direction. For a wave incident from the left, the reflection coefficient written in terms of the standard notations for the wave numbers

k_{1}={\sqrt {\frac {2mE}{\hbar ^{2}}}},k_{2}={\sqrt {\frac {2m(E-V_{0})}{\hbar ^{2}}}}

reads

R=e^{-2\pi \sigma k_{2}}{\frac {\sinh {\left({\frac {\pi \sigma }{2k_{1}}}(k_{1}-k_{2})^{2}\right)}}{\sinh {\left({\frac {\pi \sigma }{2k_{1}}}(k_{1}+k_{2})^{2}\right)}}}

References

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Lambert-W step-potential

Solution

See also

References