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arXiv:quant-ph/0206176 · Physical Review A

Quantum mechanics of a free particle on a plane with an extracted point

K. Kowalski, Krzysztof Podlaski, Jakub Rembieliński

Published Sep 27, 2002• 37 citations
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Abstract

A detailed study of a quantum free particle on a pointed plane is presented in this paper. In particular, some questions posed in the very recent paper by M. A. Cirone et al, Phys. Rev. A 65, 022101 (2002) are clarified. Namely, the topological effects related to extracting a point from a plane are indicated. The proposed results are introduced concerning self-adjoint extensions of operators describing the free particle on a pointed plane as well as the role played by discrete symmetries in the analysis of such extensions.

DOI
10.1103/physreva.66.032118

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A detailed study of a quantum free particle on a pointed plane is presented in this paper. In particular, some questions posed in the very recent paper by M. A. Cirone et al, Phys. Rev. A 65, 022101 (2002) are clarified. Namely, the topological effects related to extracting a point from a plane are indicated. The proposed results are introduced concerning self-adjoint extensions of operators describing the free particle on a pointed plane as well as the role played by discrete symmetries in the analysis of such extensions.
Quantum mechanics of a (nonrelativistic) system S localized on a topologically nontrivial manifold ℳ as its configuration space is based on a quantization method, which, in general, reflects global properties of ℳ, i.e., some of the observables of S will ‘‘feel’’ the topology: There are topological effects and inequivalent quantizations on ℳ. Some straightforward examples are given for such effects, using Borel quantization (BQ), the pointed plane as manifold ℳ, and the energy operator with harmonic potential as observable. Two topological effects exist. There are unitarily inequivalent BQ on ℳ, which are equivalent to the usual quantization on the plane with a topological potential, which has the form of a Bohm–Aharonov potential. There are different self-adjoint extensions of the energy operator for a given BQ that in some cases are related to another kind of topological potential. These effects are discussed in detail, especially the self-adjoint extensions of the energy operator. An experimental setup to verify some of the results is suggested.
In this paper, the free motion of a particle on a manifold that consists of a one-dimensional and a two-dimensional part connected in one point is discussed. The class of admissible Hamiltonians is found using the theory of self-adjoint extensions. Particular attention is paid to those Hamiltonians that allow the particle to pass through the point singularity; the reflection coefficient and other quantities characterizing scattering on the connection point are calculated. A possible application is also discussed.
We show that the intriguing localization of a free particle wave-packet is possible due to a hidden scale present in the system. Self-adjoint extensions (SAE) is responsible for introducing this scale in quantum mechanical models through the nontrivial boundary conditions. We discuss a couple of classically scale invariant free particle systems to illustrate the issue. In this context it has been shown that a free quantum particle moving on a full line may have localized wave-packet around the origin. As a generalization, it has also been shown that particles moving on a portion of a plane or on a portion of a three-dimensional space can have unusual localized wave-packet.
The notion of self-adjointness as distinct from Hermiticity is usually avoided in elementary quantum mechanics courses due to a feeling that it has nothing to do with physics or else is too mathematical. This is definitely not the case, and we use the model of a free particle restricted to motion on a line over a finite interval, to demonstrate this. For such a particle, there is one-parameter infinity of self-adjoint extensions of the momentum operator. We use one of these, corresponding to ’’antiperiodic’’ boundary conditions, to show that physical phenomena different from the usual case (periodic boundary conditions) occur. In fact, in this case, we have a model displaying many of the properties of spontaneously broken symmetry.
Chapter 4 The Quantum Plane 4.1 Position and momentum We now have the necessary machinery to treat what is probably the most fundamental example in quantum mechanics: the one-dimensional particle. Its classical analog was mentioned in Section 1.1, and there we pointed out that the phase space of this classical system can be identied with the real plane R . The phase space of the corresponding quantum system is modelled on the Hilbert space L (R), and this space, together with some associated structure, plays the role of a \\quantum" plane. As we will see in this and later chapters, the complex, topological, measure theoretic, metric, and dierentiable structures of the ordinary plane all have quantum analogs. In this section we introduce \\coordinates" and \\translations" on the quantum plane.
Making use of recent techniques in the theory of self-adjoint extensions of symmetric operators, we characterize the class of point interaction Hamiltonians in a three-dimensional bounded domain with regular boundaries. In the particular case of one point interaction acting in the center of a ball, we obtain an explicit representation of the point spectrum of the operator together with the corresponding eigenfunctions. These operators are used to build up a model system where the dynamics of a quantum particle depends on the state of a quantum bit.