Physics – Quantum Physics
Scientific paper
2011-05-02
Physics
Quantum Physics
36 pages, 5 figures
Scientific paper
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.
Al-Hashimi M. H.
Wiese Uwe-Jens
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