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Lambert-W step-potential

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

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

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

The Lambert ${\displaystyle W}$-potential is an asymmetric step of height ${\displaystyle V_{0}}$ whose steepness and asymmetry are controlled by parameter ${\displaystyle \sigma }$. 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 ${\displaystyle m}$ and energy ${\displaystyle E}$:

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

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

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

where ${\displaystyle u(z)}$ is the general solution of the scaled confluent hypergeometric equation

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

and the involved parameters are given as

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

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 ${\displaystyle W}$-potential is discussed, it is convenient to choose the general solution of the scaled confluent hypergeometric equation as

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

where ${\displaystyle c_{1,2}}$ are arbitrary constants and ${\displaystyle {}_{1}F_{1}}$ and ${\displaystyle U}$ 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

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

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

a/ Confluent hypergeometric potentials

• Quantum harmonic oscillator
• Hydrogen atom
• Morse potential
• Kratzer potential
• Inverse square root potential

b/ Hypergeometric potentials

• Pöschl–Teller potential
• Eckart potential
• Woods-Saxon potential

c/ Other potentials

• Rectangular potential barrier
• Finite potential well
• Infinite potential well
• Delta potential barrier (QM)
• Finite potential barrier (QM)

References

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