dorsal/arxiv
View SchemaExact solution and perturbation theory in a general quantum system
| Authors | An Min Wang |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0602055 |
| URL | https://arxiv.org/abs/quant-ph/0602055 |
Abstract
By splitting a Hamiltonian into two parts, using the solvability of eigenvalue problem of one part of the Hamiltonian, proving a useful identity and deducing an expansion formula of power of operator binomials, we obtain an explicit and general form of time evolution operator in the representation of solvable part of the Hamiltonian. Further we find out an exact solution of Schr\"{o}dinger equation in a general time-independent quantum system, and write down its concrete form when the solvable part of this Hamiltonian is taken as the kinetic energy term. Comparing our exact solution with the usual perturbation theory makes some features and significance of our solution clear. Moreover, through deriving out the improved forms of the zeroth, first, second and third order perturbed solutions including the partial contributions from the higher order even all order approximations, we obtain the improved transition probability. In special, we propose the revised Fermi's golden rule. Then we apply our scheme to obtain the improved forms of perturbed energy and perturbed state. In addition, we study an easy understanding example to illustrate our scheme and show its advantage. All of this implies the physical reasons and evidences why our exact solution and perturbative scheme are formally explicit, actually calculable, operationally efficient, conclusively more accurate. Therefore our exact solution and perturbative scheme can be thought of theoretical developments of quantum dynamics. Further applications of our results in quantum theory can be expected.
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"abstract": "By splitting a Hamiltonian into two parts, using the solvability of\neigenvalue problem of one part of the Hamiltonian, proving a useful identity\nand deducing an expansion formula of power of operator binomials, we obtain an\nexplicit and general form of time evolution operator in the representation of\nsolvable part of the Hamiltonian. Further we find out an exact solution of\nSchr\\\"{o}dinger equation in a general time-independent quantum system, and\nwrite down its concrete form when the solvable part of this Hamiltonian is\ntaken as the kinetic energy term. Comparing our exact solution with the usual\nperturbation theory makes some features and significance of our solution clear.\nMoreover, through deriving out the improved forms of the zeroth, first, second\nand third order perturbed solutions including the partial contributions from\nthe higher order even all order approximations, we obtain the improved\ntransition probability. In special, we propose the revised Fermi\u0027s golden rule.\nThen we apply our scheme to obtain the improved forms of perturbed energy and\nperturbed state. In addition, we study an easy understanding example to\nillustrate our scheme and show its advantage. All of this implies the physical\nreasons and evidences why our exact solution and perturbative scheme are\nformally explicit, actually calculable, operationally efficient, conclusively\nmore accurate. Therefore our exact solution and perturbative scheme can be\nthought of theoretical developments of quantum dynamics. Further applications\nof our results in quantum theory can be expected.",
"arxiv_id": "quant-ph/0602055",
"authors": [
"An Min Wang"
],
"categories": [
"quant-ph",
"math-ph",
"math.MP"
],
"title": "Exact solution and perturbation theory in a general quantum system",
"url": "https://arxiv.org/abs/quant-ph/0602055"
},
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