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

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.

36 pages, 5 figures

References in corpus (4)

• Relativistic particle in a three-dimensional box
• Casimir effect for the scalar field under Robin boundary conditions: A functional integral approach
• Waveguides with combined Dirichlet and Robin boundary conditions
• Resonant alteration of propagation in guiding structures with complex Robin parameter and its magnetic-field-induced restoration

Cited by in corpus (25)

• Self-adjoint Extensions for Confined Electrons:from a Particle in a Spherical Cavity to the Hydrogen Atom in a Sphere and on a Cone
• The Casimir effect with quantized charged spinor matter in background magnetic field
• Asymptotic Freedom, Dimensional Transmutation, and an Infra-red Conformal Fixed Point for the $\delta$-Function Potential in 1-dimensional Relativistic Quantum Mechanics
• Fermionic vacuum currents in topologically nontrivial braneworlds: Two-brane geometry
• Theory of the Robin quantum wall in a linear potential. I. Energy spectrum, polarization and quantum-information measures
• Observables in Quantum Mechanics and the Importance of Self-adjointness
• Canonical quantization on the half-line and in an interval based upon a new concept for the momentum in a space with boundaries
• Quantum confinement under Neumann condition: atomic H filled in a lattice of cavities
• Majorana Fermions in a Box
• Quantum information measures of the one-dimensional Robin quantum well
• Bouncing Wave Packets, Ehrenfest Theorem, and Uncertainty Relation based upon a new Concept for the Momentum of a Particle in a Box
• Harmonic Oscillator in a 1D or 2D Cavity with General Perfectly Reflecting Walls
• Supersymmetric Descendants of Self-Adjointly Extended Quantum Mechanical Hamiltonians
• Guiding structures with multiply connected cross-sections: evolution of propagation in external fields at complex Robin parameters
• Pressure from the vacuum of confined spinor matter
• Spontaneous spherical symmetry breaking in atomic confinement
• Effects of chiral MIT boundary conditions for a Dirac particle in a box
• Induced vacuum magnetic field in the cosmic string background
• Self-adjointness and the Casimir effect with confined quantized spinor matter
• Hot dense magnetized ultrarelativistic spinor matter in a slab
• On the chiral separation effect in a slab
• Uncertainty relations: curiosities and inconsistencies
• Contact interactions, self-adjoint extensions, and low-energy scattering
• Magnetic field in vacuum of quantum spinor matter induced by a cosmic string in three-dimensional space
• Chiral effects in magnetized quantum spinor matter in particle and astroparticle physics