Tutorial: The quantum finite square well and the Lambert W function

Abstract

We present a solution of the quantum mechanics problem of the allowable energy levels of a bound particle in a one-dimensional finite square well. The method is a geometric-analytic technique utilizing the conformal mapping w → z = we w between two complex domains. The solution of the finite square well problem can be seen to be described by the images of simple geometric shapes, lines and circles, under this map and its inverse image. The technique can also be described using the Lambert W function. One can work in either of the complex domains, thereby obtaining additional insight into the finite square well problem and its bound energy states. There are many opportunities to follow up, and we present the method in a pedagogical manner to stimulate further research in this and related avenues.

Lambert W function methods in double square well and waveguide problems

“…We wish to derive a structural equation, which is one of the relationships which must be satisfied by the energy -E in terms of the well dimensions a, L, V 0 , V 1 (and m, ÿ). To do this we manipulate equations (5) to (8) in order to eliminate the A B C , , Further simplifications are in order before we obtain the structural equation in the desired form. First, we can use our knowledge that γ is very large in problems of interest, such as an ammonia molecule model.…”

“…Roberts and Valluri [8] outline a geometric-analytic approach that uses the conformal mapping ⟶ = w z w e w to find the solutions of the allowed energy levels of a bound particle in a quantum one dimensional finite single square well (FSW). That mapping is closely related to the Lambert W function 8 . In fact, the Lambert W function is the multi-branch inverse w=W(z) of the mapping = z w e w .…”

Quantum square-well with logarithmic central spike

“…The enormous simplicity of the one-dimensional quantum square well oscillator makes it suitable for pedagogical purposes. Its analyses appear not only in conventional textbooks [3,4] but also in the less conventional studies of supersymmetric quantum systems [5,6]. The elementary nature of the square-well model found also nontrivial applications in parity times time-reversal symmetric quantum mechanics [7,8,9,10] or in certain sophisticated versions of perturbation theory [11].…”

Exponential of a matrix, a nonlinear problem, and quantum gates

“…where W is (any branch of) the Lambert W -function [2,3,4]. The Lambert W -function is defined by z = W (z) exp(W (z)) with the properties…”