From a Particle in a Box to the Uncertainty Relation in a Quantum Dot and to Reflecting Walls for Relativistic Fermions

TL;DR

This paper explores the mathematical foundations of quantum confinement, deriving generalized uncertainty relations for particles in finite domains with reflecting boundaries, and extends the analysis to relativistic fermions and complex boundary conditions.

Contribution

It introduces a unified framework for self-adjoint extensions of Hamiltonians in confined quantum systems, deriving new uncertainty relations and boundary conditions applicable to various physical contexts.

Findings

Negative energy states can arise in confined quantum systems.

A generalized uncertainty relation is established for arbitrary-shaped quantum dots.

Boundary conditions for relativistic fermions are characterized by parameter families.

Abstract

We consider a 1-parameter family of self-adjoint extensions of the Hamiltonian for a particle confined to a finite interval with perfectly reflecting boundary conditions. In some cases, one obtains negative energy states which seems to violate the Heisenberg uncertainty relation. We use this as a motivation to derive a generalized uncertainty relation valid for an arbitrarily shaped quantum dot with general perfectly reflecting walls in $d$ dimensions. In addition, a general uncertainty relation for non-Hermitean operators is derived and applied to the non-Hermitean momentum operator in a quantum dot. We also consider minimal uncertainty wave packets in this situation, and we prove that the spectrum depends monotonically on the self-adjoint extension parameter. In addition, we construct the most general boundary conditions for semiconductor heterostructures such as quantum dots, quantum wires, and quantum wells, which are characterized by a 4-parameter family of self-adjoint extensions. Finally, we consider perfectly reflecting boundary conditions for relativistic fermions confined to a finite volume or localized on a domain wall, which are characterized by a 1-parameter family of self-adjoint extensions in the $(1+1)$-d and $(2+1)$-d cases, and by a 4-parameter family in the $(3+1)$-d and $(4+1)$-d cases.