dorsal/arxiv
View SchemaGeometric Phases, Symmetries of Dynamical Invariants, and Exact Solution of the Schr\"odinger Equation
| Authors | Ali Mostafazadeh |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0101010 |
| URL | https://arxiv.org/abs/quant-ph/0101010 |
| DOI | 10.1088/0305-4470/34/32/312 |
| Journal | J.Phys.A34:6325-6338,2001 |
Abstract
We introduce the notion of the geometrically equivalent quantum systems (GEQS) as quantum systems that lead to the same geometric phases for a given complete set of initial state vectors. We give a characterization of the GEQS. These systems have a common dynamical invariant, and their Hamiltonians and evolution operators are related by symmetry transformations of the invariant. If the invariant is $T$-periodic, the corresponding class of GEQS includes a system with a $T$-periodic Hamiltonian. We apply our general results to study the classes of GEQS that include a system with a cranked Hamiltonian $H(t)=e^{-iKt}H_0e^{iKt}$. We show that the cranking operator $K$ also belongs to this class. Hence, in spite of the fact that it is time-independent, it leads to nontrivial cyclic evolutions and geometric phases. Our analysis allows for an explicit construction of a complete set of nonstationary cyclic states of any time-independent simple harmonic oscillator. The period of these cyclic states is half the characteristic period of the oscillator.
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"abstract": "We introduce the notion of the geometrically equivalent quantum systems\n(GEQS) as quantum systems that lead to the same geometric phases for a given\ncomplete set of initial state vectors. We give a characterization of the GEQS.\nThese systems have a common dynamical invariant, and their Hamiltonians and\nevolution operators are related by symmetry transformations of the invariant.\nIf the invariant is $T$-periodic, the corresponding class of GEQS includes a\nsystem with a $T$-periodic Hamiltonian. We apply our general results to study\nthe classes of GEQS that include a system with a cranked Hamiltonian\n$H(t)=e^{-iKt}H_0e^{iKt}$. We show that the cranking operator $K$ also belongs\nto this class. Hence, in spite of the fact that it is time-independent, it\nleads to nontrivial cyclic evolutions and geometric phases. Our analysis allows\nfor an explicit construction of a complete set of nonstationary cyclic states\nof any time-independent simple harmonic oscillator. The period of these cyclic\nstates is half the characteristic period of the oscillator.",
"arxiv_id": "quant-ph/0101010",
"authors": [
"Ali Mostafazadeh"
],
"categories": [
"quant-ph",
"hep-th",
"math-ph",
"math.MP"
],
"doi": "10.1088/0305-4470/34/32/312",
"journal_ref": "J.Phys.A34:6325-6338,2001",
"title": "Geometric Phases, Symmetries of Dynamical Invariants, and Exact Solution of the Schr\\\"odinger Equation",
"url": "https://arxiv.org/abs/quant-ph/0101010"
},
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