dorsal/arxiv
View SchemaTime evolution, cyclic solutions and geometric phases for the generalized time-dependent harmonic oscillator
| Authors | Qiong-Gui Lin |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0402159 |
| URL | https://arxiv.org/abs/quant-ph/0402159 |
| DOI | 10.1088/0305-4470/37/4/020 |
| Journal | J. Phys. A 37 (2004) 1345-1371 |
Abstract
The generalized time-dependent harmonic oscillator is studied. Though several approaches to the solution of this model have been available, yet a new approach is presented here, which is very suitable for the study of cyclic solutions and geometric phases. In this approach, finding the time evolution operator for the Schr\"odinger equation is reduced to solving an ordinary differential equation for a c-number vector which moves on a hyperboloid in a three-dimensional space. Cyclic solutions do not exist for all time intervals. A necessary and sufficient condition for the existence of cyclic solutions is given. There may exist some particular time interval in which all solutions with definite parity, or even all solutions, are cyclic. Criterions for the appearance of such cases are given. The known relation that the nonadiabatic geometric phase for a cyclic solution is proportional to the classical Hannay angle is reestablished. However, this is valid only for special cyclic solutions. For more general ones, the nonadiabatic geometric phase may contain an extra term. Several cases with relatively simple Hamiltonians are solved and discussed in detail. Cyclic solutions exist in most cases. The pattern of the motion, say, finite or infinite, can not be simply determined by the nature of the Hamiltonian (elliptic or hyperbolic, etc.). For a Hamiltonian with a definite nature, the motion can changes from one pattern to another, that is, some kind of phase transition may occur, if some parameter in the Hamiltonian goes through some critical value.
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"abstract": "The generalized time-dependent harmonic oscillator is studied. Though several\napproaches to the solution of this model have been available, yet a new\napproach is presented here, which is very suitable for the study of cyclic\nsolutions and geometric phases. In this approach, finding the time evolution\noperator for the Schr\\\"odinger equation is reduced to solving an ordinary\ndifferential equation for a c-number vector which moves on a hyperboloid in a\nthree-dimensional space. Cyclic solutions do not exist for all time intervals.\nA necessary and sufficient condition for the existence of cyclic solutions is\ngiven. There may exist some particular time interval in which all solutions\nwith definite parity, or even all solutions, are cyclic. Criterions for the\nappearance of such cases are given. The known relation that the nonadiabatic\ngeometric phase for a cyclic solution is proportional to the classical Hannay\nangle is reestablished. However, this is valid only for special cyclic\nsolutions. For more general ones, the nonadiabatic geometric phase may contain\nan extra term. Several cases with relatively simple Hamiltonians are solved and\ndiscussed in detail. Cyclic solutions exist in most cases. The pattern of the\nmotion, say, finite or infinite, can not be simply determined by the nature of\nthe Hamiltonian (elliptic or hyperbolic, etc.). For a Hamiltonian with a\ndefinite nature, the motion can changes from one pattern to another, that is,\nsome kind of phase transition may occur, if some parameter in the Hamiltonian\ngoes through some critical value.",
"arxiv_id": "quant-ph/0402159",
"authors": [
"Qiong-Gui Lin"
],
"categories": [
"quant-ph"
],
"doi": "10.1088/0305-4470/37/4/020",
"journal_ref": "J. Phys. A 37 (2004) 1345-1371",
"title": "Time evolution, cyclic solutions and geometric phases for the generalized time-dependent harmonic oscillator",
"url": "https://arxiv.org/abs/quant-ph/0402159"
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