dorsal/arxiv
View SchemaQuantization of the canonically conjugate pair angle and orbital angular momentum
| Authors | H. A. Kastrup |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0510234 |
| URL | https://arxiv.org/abs/quant-ph/0510234 |
| DOI | 10.1103/PhysRevA.73.052104 |
| Journal | Phys.Rev.A73:052104,2006 |
Abstract
The question how to quantize a classical system where an angle phi is one of the basic canonical variables has been controversial since the early days of quantum mechanics. The problem is that the angle is a multivalued or discontinuous variable on the corresponding phase space. The remedy is to replace phi by the smooth periodic functions cos phi and sin phi. In the case of the canonical pair (phi,l),l: orbital angular momentum (OAM), the phase space S_(phi,l) ={phi in R mod 2pi, l in R} has the global structure S^1 x R of a cylinder on which the Poisson brackets of the 3 functions cos phi, sin phi and l obey the Lie algebra of the euclidean group E(2) in the plane. This property provides the basis for the quantization of the system in terms of irreducible unitary representations of the group E(2) or of its covering groups. A crucial point is that - due to the fact that the subgroup SO(2) = S^1 is multiply connected - these representations allow for fractional OAM l = n + c, c in [0,1). Such c not 0 have already been observed in cases like the Aharonov-Bohm and the fractional quantum Hall effects and they correspond to the quasi-momenta of Bloch waves in ideal crystals. The proposal of the present paper is to look for fractional OAM in connection with the quantum optics of Laguerre-Gaussian laser modes in external magnetic fields. The quantum theory of the phase space S_(phi,l) in terms of unitary representations of E(2) allows for two types of "coherent" states the properties of which are discussed in detail: Non-holomorphic minimal uncertainty states and holomorphic ones associated with Bargmann-Segal Hilbert spaces.
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"abstract": "The question how to quantize a classical system where an angle phi is one of\nthe basic canonical variables has been controversial since the early days of\nquantum mechanics. The problem is that the angle is a multivalued or\ndiscontinuous variable on the corresponding phase space. The remedy is to\nreplace phi by the smooth periodic functions cos phi and sin phi. In the case\nof the canonical pair (phi,l),l: orbital angular momentum (OAM), the phase\nspace S_(phi,l) ={phi in R mod 2pi, l in R} has the global structure S^1 x R of\na cylinder on which the Poisson brackets of the 3 functions cos phi, sin phi\nand l obey the Lie algebra of the euclidean group E(2) in the plane. This\nproperty provides the basis for the quantization of the system in terms of\nirreducible unitary representations of the group E(2) or of its covering\ngroups. A crucial point is that - due to the fact that the subgroup SO(2) = S^1\nis multiply connected - these representations allow for fractional OAM l = n +\nc, c in [0,1). Such c not 0 have already been observed in cases like the\nAharonov-Bohm and the fractional quantum Hall effects and they correspond to\nthe quasi-momenta of Bloch waves in ideal crystals. The proposal of the present\npaper is to look for fractional OAM in connection with the quantum optics of\nLaguerre-Gaussian laser modes in external magnetic fields. The quantum theory\nof the phase space S_(phi,l) in terms of unitary representations of E(2) allows\nfor two types of \"coherent\" states the properties of which are discussed in\ndetail: Non-holomorphic minimal uncertainty states and holomorphic ones\nassociated with Bargmann-Segal Hilbert spaces.",
"arxiv_id": "quant-ph/0510234",
"authors": [
"H. A. Kastrup"
],
"categories": [
"quant-ph",
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"physics.optics"
],
"doi": "10.1103/PhysRevA.73.052104",
"journal_ref": "Phys.Rev.A73:052104,2006",
"title": "Quantization of the canonically conjugate pair angle and orbital angular momentum",
"url": "https://arxiv.org/abs/quant-ph/0510234"
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