dorsal/arxiv
View SchemaQuantum mechanics in general quantum systems (I): Exact solution
| Authors | An Min Wang |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0611216 |
| URL | https://arxiv.org/abs/quant-ph/0611216 |
Abstract
Starting from our idea of combining the Feynman path integral spirit and the Dyson series kernel, we find an explicit and general form of time evolution operator that is a $c$-number function and a power series of perturbation including all order approximations in the unperturbed Hamiltonian representation. Based on it, we obtain an exact solution of the Schr\"{o}dinger equation in general quantum systems independent of time. Comparison of our exact solution with the existed perturbation theory makes some features and significance of our exact solution clear. The conclusions expressly indicate that our exact solution is obviously consistent with the usual time-independent perturbation theory at any order approximation, it explicitly calculates out the expanding coefficients of the unperturbed state in the non-perturbation method, and it fully solves the recurrence equation of the expansion coefficients of final state in the unperturbed Hamiltonian representation from a view of time-dependent perturbation theory. At the same time, the exact solution of the von Neumann equation is also given. Our results can be thought of as theoretical developments of quantum dynamics, and are helpful for understanding the dynamical behavior and related subjects of general quantum systems in both theory and application. Our exact solution, together with its sequence studies on perturbation theory [An Min Wang, quant-ph/0611217] and open system dynamics [An Min Wang, quant-ph/0601051] can be used to establish the foundation of theoretical formulism of quantum mechanics in general quantum systems. Further applications of our exact solution to quantum theory can be expected.
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"abstract": "Starting from our idea of combining the Feynman path integral spirit and the\nDyson series kernel, we find an explicit and general form of time evolution\noperator that is a $c$-number function and a power series of perturbation\nincluding all order approximations in the unperturbed Hamiltonian\nrepresentation. Based on it, we obtain an exact solution of the Schr\\\"{o}dinger\nequation in general quantum systems independent of time. Comparison of our\nexact solution with the existed perturbation theory makes some features and\nsignificance of our exact solution clear. The conclusions expressly indicate\nthat our exact solution is obviously consistent with the usual time-independent\nperturbation theory at any order approximation, it explicitly calculates out\nthe expanding coefficients of the unperturbed state in the non-perturbation\nmethod, and it fully solves the recurrence equation of the expansion\ncoefficients of final state in the unperturbed Hamiltonian representation from\na view of time-dependent perturbation theory. At the same time, the exact\nsolution of the von Neumann equation is also given. Our results can be thought\nof as theoretical developments of quantum dynamics, and are helpful for\nunderstanding the dynamical behavior and related subjects of general quantum\nsystems in both theory and application. Our exact solution, together with its\nsequence studies on perturbation theory [An Min Wang, quant-ph/0611217] and\nopen system dynamics [An Min Wang, quant-ph/0601051] can be used to establish\nthe foundation of theoretical formulism of quantum mechanics in general quantum\nsystems. Further applications of our exact solution to quantum theory can be\nexpected.",
"arxiv_id": "quant-ph/0611216",
"authors": [
"An Min Wang"
],
"categories": [
"quant-ph",
"math-ph",
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"title": "Quantum mechanics in general quantum systems (I): Exact solution",
"url": "https://arxiv.org/abs/quant-ph/0611216"
},
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