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Exact Solutions for Compact Support Parabolic and Landau Barriers

Peter Collas, David Klein

TL;DR

This work provides exact analytic solutions to the one-dimensional Schrödinger equation for compactly supported barriers of parabolic and Landau–Lifshitz types, including combinations thereof. By mapping the interior barrier problems to known special functions—confluent hypergeometric functions for the parabolic case and associated Legendre functions for the Landau–Lifshitz case—the authors derive closed-form expressions for reflection and transmission coefficients and for dwell times, and they demonstrate a resonant quasi-bound state in a double-barrier configuration. The results hinge on precise boundary-matching within compact support, enabling exact, differentiable wavefunctions and enabling construction of multi-barrier and mixed-barrier configurations with guaranteed continuity and smoothness. Collectively, the study advances analytic tunneling analyses for smooth, finite-range potentials and provides a toolkit for exact scattering in engineered compact barriers.

Abstract

We derive exact solutions to the one-dimensional Schrödinger equation for compact support parabolic and hyperbolic secant potential barriers, along with combinations of these types of potential barriers. We give the expressions for transmission and reflection coefficients and calculate some dwell times of interest

Exact Solutions for Compact Support Parabolic and Landau Barriers

TL;DR

This work provides exact analytic solutions to the one-dimensional Schrödinger equation for compactly supported barriers of parabolic and Landau–Lifshitz types, including combinations thereof. By mapping the interior barrier problems to known special functions—confluent hypergeometric functions for the parabolic case and associated Legendre functions for the Landau–Lifshitz case—the authors derive closed-form expressions for reflection and transmission coefficients and for dwell times, and they demonstrate a resonant quasi-bound state in a double-barrier configuration. The results hinge on precise boundary-matching within compact support, enabling exact, differentiable wavefunctions and enabling construction of multi-barrier and mixed-barrier configurations with guaranteed continuity and smoothness. Collectively, the study advances analytic tunneling analyses for smooth, finite-range potentials and provides a toolkit for exact scattering in engineered compact barriers.

Abstract

We derive exact solutions to the one-dimensional Schrödinger equation for compact support parabolic and hyperbolic secant potential barriers, along with combinations of these types of potential barriers. We give the expressions for transmission and reflection coefficients and calculate some dwell times of interest
Paper Structure (11 sections, 2 theorems, 112 equations, 9 figures)

This paper contains 11 sections, 2 theorems, 112 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Parabolic barrier with compact support
  3. Reflection and transmission coefficients
  4. Multiple parabolic barriers
  5. Quasi-bound states and dwell times
  6. The Landau and Lifshitz barrier and its compact support version
  7. Summary
  8. Coordinate transformation
  9. Power series representation of $w_e$ and $w_o$
  10. Dwell time and probability current
  11. Inserting the $m$ and $\hbar$ back in the solutions

Key Result

Proposition 1

Let $\sigma>0$, assume that $\lambda$ is real and that $V(x)$ is a solution of the differential equation Then the function is a solution to Conversely, if $w(z)$ is a solution to where the parameter $a$ is real, then the function is a solution to

Figures (9)

  • Figure 1: The parabolic barrier $U(x)$ with $\alpha=1/2,\;U_0=1$. The dashed vertical lines show the locations of the two turning points $x_0\approx 0.35$ for an incoming particle with kinetic energy $k^2/2=1/2$ with the convention $\hbar=m=1$.
  • Figure 2: A double parabolic barrier. For the parabola on the left, $U(\alpha,\gamma,U_0,x)=U(1,-2,1,x)$ and on the right $U(\alpha,\gamma,U_0,x)=U(1,2,2,x)$.
  • Figure 3: A double barrier with turning points for a particle with energy $E<U_0$ at $a, b, c, d$.
  • Figure 4: We show a plot of $|\varphi|^2$ for the double barrier in Fig. \ref{['doubleBsym']} using the values in Eqs, \ref{['F1SIa']} and \ref{['F1SIb']} for the parameters and $\alpha=-\gamma_1$ for the barrier on the left and $\alpha=\gamma_2$ for the barrier on the right. The incident particle's energy is $E=0.02$h.
  • Figure 5: This is again a plot of $|\varphi(x)|^2$ for the double barrier in Fig. \ref{['doubleBsym']} using the values in Eqs, \ref{['F1SIa']} and \ref{['F1SIb']} for the parameters and $\alpha=-\gamma_1$ for the barrier on the left and $\alpha=\gamma_2$ for the barrier on the right. The incident particle's energy is $E=0.06115146$ h which is the energy of the quasi-bound state.
  • ...and 4 more figures

Theorems & Definitions (9)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Remark 5