2011/05/02 by M.H. Al-Hashimi, M. H. Al-Hashimi, U. -J. Wiese +1
Computer Science · Mathematics · Physics and Astronomy · #Boundary (topology) #Boundary value problem #Domain (mathematical analysis) #Fermion #Momentum (technical analysis) #Operator (biology) #Quantum #Quantum Information and Cryptography #Quantum Mechanics and Non-Hermitian Physics #Quantum dot #Spectral Theory in Mathematical Physics #Uncertainty principle #hep-th #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1016/j.aop.2011.05.003
published as Annals of Physics 327 (2012) 2742-2759 · 36 pages, 5 figures
arxiv created 2011/05/02 · openalex publication_date 2011/05/26 · arxiv updated 2015/05/28 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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.