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From a Particle in a Box to the Uncertainty Relation in a Quantum Dot and to Reflecting Walls for Relativistic Fermions
- Source :
- Annals of Physics 327 (2012) 2742-2759
- Publication Year :
- 2011
-
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.<br />Comment: 36 pages, 5 figures
- Subjects :
- Quantum Physics
High Energy Physics - Theory
Mathematical Physics
Subjects
Details
- Database :
- arXiv
- Journal :
- Annals of Physics 327 (2012) 2742-2759
- Publication Type :
- Report
- Accession number :
- edsarx.1105.0391
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1016/j.aop.2011.05.003