Lambert-W step-potential

The Lambert-W step-potential 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 potentials) in terms of the confluent hypergeometric functions. 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

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

See also

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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