dorsal/arxiv
View SchemaSolutions of the Schr\"{o}dinger equation for the time-dependent linear potential
| Authors | Jian Qi Shen |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0310179 |
| URL | https://arxiv.org/abs/quant-ph/0310179 |
Abstract
By making use of the Lewis-Riesenfeld invariant theory, the solution of the Schr\"{o}dinger equation for the time-dependent linear potential corresponding to the quadratic-form Lewis-Riesenfeld invariant $I_{\rm q}(t)$ is obtained in the present paper. It is emphasized that in order to obtain the general solutions of the time-dependent Schr\"{o}dinger equation, one should first find the complete set of Lewis-Riesenfeld invariants. For the present quantum system with a time-dependent linear potential, the linear $I_{\rm l}(t)$ and quadratic $I_{\rm q}(t)$ (where the latter $I_{\rm q}(t)$ cannot be written as the squared of the former $I_{\rm l}(t)$, {\it i.e.}, the relation $I_{\rm q}(t)= cI_{\rm l}^{2}(t)$ does not hold true always) will form a complete set of Lewis-Riesenfeld invariants. It is also shown that the solution obtained by Bekkar {\it et al.} more recently is the one corresponding to the linear $I_{\rm l}(t)$, one of the invariants that form the complete set. In addition, we discuss some related topics regarding the comment [Phys. Rev. A {\bf 68}, 016101 (2003)] of Bekkar {\it et al.} on Guedes's work [Phys. Rev. A {\bf 63}, 034102 (2001)] and Guedes's corresponding reply [Phys. Rev. A {\bf 68}, 016102 (2003)].
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"abstract": "By making use of the Lewis-Riesenfeld invariant theory, the solution of the\nSchr\\\"{o}dinger equation for the time-dependent linear potential corresponding\nto the quadratic-form Lewis-Riesenfeld invariant $I_{\\rm q}(t)$ is obtained in\nthe present paper. It is emphasized that in order to obtain the general\nsolutions of the time-dependent Schr\\\"{o}dinger equation, one should first find\nthe complete set of Lewis-Riesenfeld invariants. For the present quantum system\nwith a time-dependent linear potential, the linear $I_{\\rm l}(t)$ and quadratic\n$I_{\\rm q}(t)$ (where the latter $I_{\\rm q}(t)$ cannot be written as the\nsquared of the former $I_{\\rm l}(t)$, {\\it i.e.}, the relation $I_{\\rm q}(t)=\ncI_{\\rm l}^{2}(t)$ does not hold true always) will form a complete set of\nLewis-Riesenfeld invariants. It is also shown that the solution obtained by\nBekkar {\\it et al.} more recently is the one corresponding to the linear\n$I_{\\rm l}(t)$, one of the invariants that form the complete set. In addition,\nwe discuss some related topics regarding the comment [Phys. Rev. A {\\bf 68},\n016101 (2003)] of Bekkar {\\it et al.} on Guedes\u0027s work [Phys. Rev. A {\\bf 63},\n034102 (2001)] and Guedes\u0027s corresponding reply [Phys. Rev. A {\\bf 68}, 016102\n(2003)].",
"arxiv_id": "quant-ph/0310179",
"authors": [
"Jian Qi Shen"
],
"categories": [
"quant-ph"
],
"title": "Solutions of the Schr\\\"{o}dinger equation for the time-dependent linear potential",
"url": "https://arxiv.org/abs/quant-ph/0310179"
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