Square well/barrier resonances in the potentiodynamic plane

Highlights

• A fully completed study of resonances and anti-resonances of a square potential; both well and barrier cases are considered.

• A detailed analysis of how bound states of the well evolve from the original pairs of the resonant and antiresonant states.

• An illustration of the behavior of the resonance/antiresonance flows in the complex wave number and potentiodynamic planes.

Abstract

The complex wave number plane equips us with an elegant mathematical construct which can be used to display and classify the different types of states which arise from solution of the time-independent Schrödinger equation for a quantum mechanical potential. The complex wave number plane is also useful for tracking the trajectories of these solutions as the potential is perturbed in some way, often resulting in profound dynamical structure. In this work we propose an alternative coordinate system, which we call the potentiodynamic plane, which has the useful property that the trajectories stay bounded, and apply this to the square well/barrier potential to reveal some new insights.

Introduction

Resonances and resonant phenomena appear in many branches of mathematics, physics, and engineering. They are especially important in the fields of semiconductor nanostructures, microcavity lasers, and biosensing, see Refs. [1], [2], [3], [4].

In quantum mechanics the concept of a resonant state (resonance) is a natural extension of the idea of a bound state. It was introduced in Refs. [5], [6] and it is sometimes referred as the Gamow vector. The coordinate parts of resonant states ${\psi }_n(\mathbf{r})$ are solutions of the time-independent Schrödinger equation,

$$[-\mathrm{\Delta }+V(\mathbf{r})]\psi (\mathbf{r})=\mathcal{E}\psi (\mathbf{r})$$

with complex eigenvalues

$${\mathcal{E}}_n=E_n-\frac{1}{2}i{\mathrm{\Gamma }}_n,{\mathrm{\Gamma }}_n>0,$$

on which the outgoing boundary conditions are imposed, see Eqs. (5), (6) for details. The overall wave functions ${\mathrm{\Psi }}_n(\mathbf{r},t)$ of resonant states decay in time in accordance with

$${\mathrm{\Psi }}_n(\mathbf{r},t)={\psi }_n(\mathbf{r})\exp (-i{E}_{n}t)\exp (-{\mathrm{\Gamma }}_n/2t).$$

In obeyance with time-reversal invariance, the above resonant states should be accompanied by another set of solutions of Eq. (1), which have the form ${\psi }_{-n}(\mathbf{r})={\psi }_n^{⁎}(\mathbf{r})$, with the corresponding eigenvalues ${\mathcal{E}}_{-n}={\mathcal{E}}_n^{⁎}$, and the overall wave functions ${\mathrm{\Psi }}_{-n}(\mathbf{r},t)={\mathrm{\Psi }}_n^{⁎}(\mathbf{r},-t)$, i.e.

$${\mathcal{E}}_{-n}=E_n+\frac{1}{2}i{\mathrm{\Gamma }}_n,{\mathrm{\Gamma }}_n>0,{\mathrm{\Psi }}_{-n}(\mathbf{r},t)={\psi }_n^{⁎}(\mathbf{r})\exp (-i{E}_{n}t)\exp ({\mathrm{\Gamma }}_n/2t).$$

These states grow in time and are often referred as antiresonant states (antiresonances). In the above and further in this paper, Rydberg units, i.e. $ħ=1$ and $m=1/2$, where m is the particle mass (for example, the effective mass of an electron in a semiconductor) are used.

As mentioned, the states given by Eqs. (3) are indeed resonant states, provided that ${\psi }_n(\mathbf{r})$ satisfy the outgoing boundary conditions. To impose these conditions, it is convenient to characterize the resonant states by the complex eigenwavenumber $k_n={\alpha }_n+i{\beta }_n$, where ${\alpha }_n=\mathrm{Re}(k_n)$ and ${\beta }_n=\mathrm{Im}(k_n)$, related to the complex energy ${\mathcal{E}}_n$ as

$${\mathcal{E}}_n=k_n^2={\alpha }_n^2-{\beta }_n^2+2i{\alpha }_n{\beta }_n.$$

The outgoing boundary conditions select from a pool of functions ${\psi }_n(\mathbf{r})$, satisfying Eq. (1), only those obeying

$${\psi }_n(\mathbf{r})\sim e^{i{\alpha }_{n}r}{e}^{-{\beta }_{n}r},{\alpha }_n>0,r\to \infty .$$

The resonant states ${\psi }_n(\mathbf{r})$ have then ${\beta }_n<0$ as ${\mathrm{\Gamma }}_n>0$ in Eq. (2), i.e. their eigenwavenumbers $k_n$ are located in the fourth quadrant of the complex wave number plane. The states are not orthogonal and not normalizable in the usual way, see Refs. [7], [8], [9].

The antiresonant states ${\psi }_{-n}(\mathbf{r})={\psi }_n^{⁎}(\mathbf{r})$ satisfy the asymptotic condition

$${\psi }_{-n}(\mathbf{r})\sim e^{-i{\alpha }_{n}r}{e}^{-{\beta }_{n}r},r\to \infty .$$

Their eigenwavenumbers $k_{-n}={\alpha }_{-n}+i{\beta }_{-n}=-{\alpha }_n+i{\beta }_n$ are located in the third quadrant of the complex wave number plane.

A special case ${\alpha }_n=0$ corresponds to real negative energies ${\mathcal{E}}_n=E_n=-{\beta }_n^2<0$. It includes either the bound states with ${\beta }_n>0$ and coordinate wave functions ${\psi }_n(\mathbf{r})$ vanishing far from the potential, or antibound states with ${\beta }_n<0$ and purely growing functions ${\psi }_n(\mathbf{r})$ outside of the potential.

As one can see, we defined the resonant/antiresonant states as explicit solutions of the Schrödinger equation with complex eigenvalues and eigenfunctions subject to the boundary conditions given by Eqs. (6), (7). This is an approach illustrated in Refs. [8], [9], [10], [11]. Another way to introduce these states is to associate them with the poles of the outgoing Green's function, see Refs. [8], [9], or with the poles of the scattering matrix S, see Refs. [12], [13].

The purpose of this paper is to understand how the resonant/antiresonant states evolve, develop and possibly transform into bound states of a quantum mechanical system. This is an interesting and useful exercise, especially in the context of a newly-adopted resonant-state expansion method, see Refs. [14], [15], [16], [17]. The complex wave number plane provides one option of a natural setting in which to pursue this, see Refs. [11], [12], [13], [18] for the 1D quantum problem and Refs. [19], [20] for the 2D quantum problem. In this paper, however, using a square well/barrier as an example, we present an alternative complex plane, which we call the potentiodynamic wave number coordinate system. In this plane we can learn more about the behavior of the full spectrum of eigenstates of Eq. (1), especially in the limit of large potentials. As a result, the square well/barrier potential system will serve as a proof of concept for potentiodynamic coordinates before applying the technique in a more complicated setting where special functions often feature, obfuscating the details.