A simple quantum dot: numerical and variational solutions

TL;DR

This study analyzes a geometry-induced bound state in a simple crossed-trough quantum dot, where no classical bound state exists. It benchmarks several numerical strategies—matrix mechanics with a 2D box basis, finite-difference discretization, and the analytic mode-matching method—alongside a tight-binding toy model and a compact variational wave function. The mode-matching approach delivers the most accurate ground-state energy, $E = 0.659606\,E_t$, while a minimal variational form achieves $E = 0.6812\,E_t$, illustrating how geometry alone can sustain binding and how numerical methods can yield compact analytical insights. The results provide an educational, versatile toolkit for undergraduates to address nontrivial quantum geometries and to connect numerical results with variational and analytical perspectives.

Abstract

We describe a simple quantum dot that consists of two crossed troughs. As such there is no potential well; nonetheless this geometry gives rise to a bound state, centred around the point at which these troughs cross one another. In this paper we review existing numerical methods to solve this problem, and highlight one which we feel is particularly elegant and, in this case, provides the most accurate solution to the problem. The bound state is well-contained on the scale of the trough width, and yields a bound state energy of $0.659606$ in units of the minimum continuum state energy. This method also motivates a simple variational solution which yields the lowest energy known to date ($0.6812$ in the same units) to arise out of an analytical variational solution.

Figures (8)

• Figure 1: Schematic of the crossed troughs of width $a$ (shaded in white), with grey-shaded regions with a very large or infinite potential ($V_0$) that serves to confine the particle to the troughs. No bound state is expected to exist, classically.
• Figure 2: Ground state wave function, computed with Eq. \ref{['MMexpansion']} for $a/L=0.15$ and ${\rm v}_0 \equiv V_0/E_{\rm t} = 500$. Note that it is self-evidently bounded near the origin. We have confirmed that similar plots are obtained as $L/a$ is increased and for an enormous range of $V_0$ values.
• Figure 3: Bound state energy versus $1/\sqrt{\rm {v}_0}$ for $a/L = 0.2$ (unfilled square symbols) and for $a/L=0.1$ (filled circular symbols). Note that the lower set of symbols shows $\epsilon \equiv E/E_{\rm t}$, while the upper set of symbols is a better measure of the boundedness, and plots $E/E_{\rm t}^{V_0}$. The actual value of ${\rm v}_0 \equiv V_0/E_{\rm t}$ for each symbol is shown across the top of the graph. Note that the variability of $E/E_{\rm t}^{V_0}$ with $V_0$ is a lot less than it is for $\epsilon$. When the width of the trough is this small there is hardly any dependence on trough width. The inset (red curve) shows the normalized threshold energy, $E_{\rm t}^{V_0}/E_{\rm t}$ for a bound state as a function of ${\rm v}_0$. The thin black curve gives the first-order analytical expansion (see Eq. (\ref{['expansion']}) in Appendix \ref{['matrixappendix']}).
• Figure 4: A sketch of the crossed trough geometry, with labels for the individual regions on which we solve the Schrödinger equation. In our treatment, the arms (regions A, B, C, and D) extend to infinity.
• Figure 5: Ground state wave function for $L\rightarrow \infty$ and $V_0\rightarrow \infty$ (solid red curves, obtained via mode-matching) and for ${\rm v}_0\equiv V_0/E_{\rm t}=500$ and $a/L = 0.15$ (dashed black curves, obtained via matrix mechanics). Three representative slices of constant $y$ are shown for comparison. For the finite $V_0$ case, the wave function leaks out of the troughs, which is especially apparent in the $y_0=a/2$ curve. This makes the wave function less sharply peaked in the centre, most evident in the $y_0 = 0$ slice.