Square well/barrier resonances in the potentiodynamic plane
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 are solutions of the time-independent Schrödinger equation, with complex eigenvalues on which the outgoing boundary conditions are imposed, see Eqs. (5), (6) for details. The overall wave functions of resonant states decay in time in accordance with
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 , with the corresponding eigenvalues , and the overall wave functions , i.e. 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. and , 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 satisfy the outgoing boundary conditions. To impose these conditions, it is convenient to characterize the resonant states by the complex eigenwavenumber , where and , related to the complex energy as The outgoing boundary conditions select from a pool of functions , satisfying Eq. (1), only those obeying The resonant states have then as in Eq. (2), i.e. their eigenwavenumbers 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 satisfy the asymptotic condition Their eigenwavenumbers are located in the third quadrant of the complex wave number plane.
A special case corresponds to real negative energies . It includes either the bound states with and coordinate wave functions vanishing far from the potential, or antibound states with and purely growing functions 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.
Section snippets
Resonant states of square well potential
The square well potential of width 2a is defined as where . Eq. (1) takes the form where all three parameters involved, , k, and κ, have in general complex values. A general solution of Eq. (9) can be represented as The boundary conditions given by Eqs. (6), (7), where for the 1D case, r should be replaced by z,
Flows
In this section, first, we numerically solve Eq. (11) to find the locations of the resonances/antiresonances in the complex wave number plane and then follow their trajectories in response to varying the depth of the well. The trajectories traced out in this procedure are known as flows, and a plot of all flows we call the flow portrait. As mentioned in the introduction, in this paper we also intend to reveal new features of flow behavior using an alternative (potentiodynamic) complex
Square barrier
To complete considerations of 1D square potentials, we also treat the case of a 1D square barrier, which is actually much simpler to interpret. The square barrier potential of width 2a is defined as To find and analyze its resonances one needs to follow all steps in Section 2. In particular, Eqs. (9) - (17) are also applicable to the barrier case, if the potentiodynamic parameter κ is redefined as
The overall flow portrait in ka coordinates is shown in
Conclusions
In this paper we studied the flows of resonances (antiresonances) for the square well potential. In particular, beginning with a very shallow well, it was illustrated how these states appear, develop, and possibly transform into bound states in response to increasing the depth of the well. Along with the complex wave number plane, the results were illustrated in a newly introduced potentiodynamic complex plane. We have shown that the potentiodynamic plane is a useful new addition to the tools
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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