A simple quantum dot: numerical and variational solutions

Technical Summary: A Simple Quantum Dot: Numerical and Variational Solutions

Problem Statement
This paper investigates a two-dimensional quantum system consisting of two infinitely long, perpendicular troughs of zero potential width $a$, intersecting to form a cross. The regions outside these troughs possess a potential $V_0$ that is either very large or infinite. Classically, a particle in this geometry possesses no bound state; it would simply propagate indefinitely along one of the arms. However, the quantum mechanical problem exhibits a bound state centered at the intersection, a phenomenon previously demonstrated via the variational principle in standard textbooks (e.g., Griffiths). The authors propose this system as a compelling case study for undergraduate quantum mechanics, offering a problem where a bound state exists despite the absence of a traditional potential well, and which is amenable to solution via multiple computational techniques.

Methodology
The authors employ and compare three distinct numerical methods to solve the time-independent Schrödinger equation for this geometry, alongside a tight-binding effective model and a variational analysis:

1. Matrix Mechanics: The potential is embedded within a large, finite two-dimensional infinite square well (box) of side length $L$. The wave function is expanded in terms of the known eigenstates of this box. The Hamiltonian is projected onto this basis to form a matrix equation, which is then diagonalized. This method requires $V_0 < \infty$ and $L \gg a$ to ensure the artificial boundaries do not influence the bound state.
2. Finite Differences: The domain is discretized into a grid, and the Laplacian operator is approximated using finite-difference formulas. This approach assumes $V_0 \to \infty$, confining the particle strictly to the troughs. The resulting sparse matrix is diagonalized to find eigenvalues.
3. Mode Matching: The geometry is divided into regions (arms and a central intersection). Analytic solutions (Fourier modes) are constructed for each region. By enforcing the continuity of the wave function and its derivatives across the boundaries, a system of linear equations for the expansion coefficients is derived. The bound state energy is found by determining the values of energy for which the determinant of the coefficient matrix vanishes.
4. Tight-Binding Model: An effective Hamiltonian describing a particle hopping on two intersecting one-dimensional atomic chains is analyzed to demonstrate that the bound state physics emerges even in this simplified, discrete limit.
5. Variational Approach: A trial wave function is constructed based on the simplest truncation ($N=1$) of the mode-matching expansion. The energy expectation value is minimized with respect to a variational parameter.

Key Results

• Ground State Energy: The most accurate solution is obtained via the mode-matching method, yielding a dimensionless ground state energy of $\epsilon \equiv E/E_t = 0.659606$, where $E_t = \hbar^2\pi^2/(2m_0a^2)$ is the threshold energy for a particle in a single arm.
• Method Comparison:
  • Matrix Mechanics: Successfully reproduces the bound state and allows for the study of finite barrier heights ($V_0$). The authors note that the degree of boundedness is surprisingly insensitive to the barrier height. However, the method requires careful extrapolation to the limit $V_0 \to \infty$ and $L \to \infty$.
  • Finite Differences: Provides good results with minimal effort but exhibits slow convergence with mesh size. It is less flexible regarding non-rectangular geometries or finite barriers.
  • Mode Matching: Demonstrates the fastest convergence. Truncating the series at just one term ($N=1$) yields an energy of $\epsilon \approx 0.6812$ (within $\sim 3\%$ of the exact value), while $N=15$ reduces the error to $10^{-3}$.
• Tight-Binding Validation: The tight-binding model yields a ground state energy of $E = -2.3094t$, which corresponds to $\epsilon \approx 0.6852$ when mapped to the continuum limit. This confirms that the bound state is a robust geometric feature, persisting even in crude approximations.
• Variational Solution: The variational wave function derived from the $N=1$ mode-matching truncation yields $\epsilon = 0.6812$. This result is significantly lower (better) than the variational solution presented in standard textbooks ($\epsilon = 0.7337$), demonstrating that a simple analytical form motivated by numerical insights can outperform physically motivated guesses.

Significance and Claims
The paper positions this work as an educational resource that bridges the gap between standard textbook problems and advanced computational techniques. The authors claim that:

• The crossed-trough problem serves as an ideal pedagogical tool to illustrate the existence of quantum bound states in classically unbound systems.
• The mode-matching method is the most elegant and accurate technique for this specific problem, offering rapid convergence and a direct path to an analytical variational wave function.
• The derived variational wave function ($\epsilon = 0.6812$) represents the lowest energy known to date arising from an analytical variational solution for this problem, outperforming previous textbook examples.
• The study highlights how numerical results (specifically the mode-matching expansion) can inspire simple, accurate analytical approximations.

The authors conclude that while matrix mechanics offers flexibility for varying potential shapes, mode matching is superior for the specific geometry of infinite perpendicular troughs. They suggest future extensions could involve non-perpendicular troughs or three-dimensional tubes, which might require the flexibility of the matrix mechanics approach.