# 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

- .

where is the Lambert function also known as the product logarithm. This is an implicitly elementary function that resolves the equation .

The Lambert -potential is an asymmetric step of height whose steepness and asymmetry are controlled by parameter . 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[edit]

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

- ,

for the Lambert -barrier for arbitrary and is written as

- ,

where is the general solution of the scaled confluent hypergeometric equation

and the involved parameters are given as

- .

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 -potential is discussed, it is convenient to choose the general solution of the scaled confluent hypergeometric equation as

- ,

where are arbitrary constants and and 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

reads

## See also[edit]

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[edit]

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- Quantum mechanics
- Schrödinger equation
- Quantum models