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Lambert-W step-potential

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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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