dorsal/arxiv
View SchemaGeometric Quantum Mechanics
| Authors | Dorje C. Brody, Lane P. Hughston |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9906086 |
| URL | https://arxiv.org/abs/quant-ph/9906086 |
| DOI | 10.1016/S0393-0440(00)00052-8 |
| Journal | J.Geom.Phys. 38 (2001) 19-53 |
Abstract
The manifold of pure quantum states is a complex projective space endowed with the unitary-invariant geometry of Fubini and Study. According to the principles of geometric quantum mechanics, the detailed physical characteristics of a given quantum system can be represented by specific geometrical features that are selected and preferentially identified in this complex manifold. Here we construct a number of examples of such geometrical features as they arise in the state spaces for spin-1/2, spin-1, and spin-3/2 systems, and for pairs of spin-1/2 systems. A study is undertaken on the geometry of entangled states, and a natural measure is assigned to the degree of entanglement of a given state for a general multi-particle system. The properties of this measure are analysed for the entangled states of a pair of spin-1/2 particles. With the specification of a quantum Hamiltonian, the resulting Schrodinger trajectory induces a Killing field, which is quasiergodic on a toroidal subspace of the energy surface. When the dynamical trajectory is lifted orthogonally to Hilbert space, it induces a geometric phase shift on the wave function. The uncertainty of an observable in a given state is the length of the gradient vector of the level surface of the expectation of the observable in that state, a fact that allows us to calculate higher order corrections to the Heisenberg relations. A general mixed state is determined by a probability density function on the state space, for which the associated first moment is the density matrix. The advantage of a general state is in its applicability in various attempts to go beyond the standard quantum theory.
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"abstract": "The manifold of pure quantum states is a complex projective space endowed\nwith the unitary-invariant geometry of Fubini and Study. According to the\nprinciples of geometric quantum mechanics, the detailed physical\ncharacteristics of a given quantum system can be represented by specific\ngeometrical features that are selected and preferentially identified in this\ncomplex manifold. Here we construct a number of examples of such geometrical\nfeatures as they arise in the state spaces for spin-1/2, spin-1, and spin-3/2\nsystems, and for pairs of spin-1/2 systems. A study is undertaken on the\ngeometry of entangled states, and a natural measure is assigned to the degree\nof entanglement of a given state for a general multi-particle system. The\nproperties of this measure are analysed for the entangled states of a pair of\nspin-1/2 particles. With the specification of a quantum Hamiltonian, the\nresulting Schrodinger trajectory induces a Killing field, which is quasiergodic\non a toroidal subspace of the energy surface. When the dynamical trajectory is\nlifted orthogonally to Hilbert space, it induces a geometric phase shift on the\nwave function. The uncertainty of an observable in a given state is the length\nof the gradient vector of the level surface of the expectation of the\nobservable in that state, a fact that allows us to calculate higher order\ncorrections to the Heisenberg relations. A general mixed state is determined by\na probability density function on the state space, for which the associated\nfirst moment is the density matrix. The advantage of a general state is in its\napplicability in various attempts to go beyond the standard quantum theory.",
"arxiv_id": "quant-ph/9906086",
"authors": [
"Dorje C. Brody",
"Lane P. Hughston"
],
"categories": [
"quant-ph"
],
"doi": "10.1016/S0393-0440(00)00052-8",
"journal_ref": "J.Geom.Phys. 38 (2001) 19-53",
"title": "Geometric Quantum Mechanics",
"url": "https://arxiv.org/abs/quant-ph/9906086"
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