dorsal/arxiv
View SchemaSewing sound quantum flesh onto classical bones
| Authors | Edward D. Davis |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0404102 |
| URL | https://arxiv.org/abs/quant-ph/0404102 |
| DOI | 10.1103/PhysRevA.70.032101 |
| Journal | Phys. Rev. A 70, 032101 (2004) |
Abstract
Semiclassical transformation theory implies an integral representation for stationary-state wave functions $\psi_m(q)$ in terms of angle-action variables ($\theta,J$). It is a particular solution of Schr\"{o}dinger's time-independent equation when terms of order $\hbar^2$ and higher are omitted, but the pre-exponential factor $A(q,\theta)$ in the integrand of this integral representation does not possess the correct dependence on $q$. The origin of the problem is identified: the standard unitarity condition invoked in semiclassical transformation theory does not fix adequately in $A(q,\theta)$ a factor which is a function of the action $J$ written in terms of $q$ and $\theta$. A prescription for an improved choice of this factor, based on succesfully reproducing the leading behaviour of wave functions in the vicinity of potential minima, is outlined. Exact evaluation of the modified integral representation via the Residue Theorem is possible. It yields wave functions which are not, in general, orthogonal. However, closed-form results obtained after Gram-Schmidt orthogonalization bear a striking resemblance to the exact analytical expressions for the stationary-state wave functions of the various potential models considered (namely, a P\"{o}schl-Teller oscillator and the Morse oscillator).
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"abstract": "Semiclassical transformation theory implies an integral representation for\nstationary-state wave functions $\\psi_m(q)$ in terms of angle-action variables\n($\\theta,J$). It is a particular solution of Schr\\\"{o}dinger\u0027s time-independent\nequation when terms of order $\\hbar^2$ and higher are omitted, but the\npre-exponential factor $A(q,\\theta)$ in the integrand of this integral\nrepresentation does not possess the correct dependence on $q$. The origin of\nthe problem is identified: the standard unitarity condition invoked in\nsemiclassical transformation theory does not fix adequately in $A(q,\\theta)$ a\nfactor which is a function of the action $J$ written in terms of $q$ and\n$\\theta$. A prescription for an improved choice of this factor, based on\nsuccesfully reproducing the leading behaviour of wave functions in the vicinity\nof potential minima, is outlined. Exact evaluation of the modified integral\nrepresentation via the Residue Theorem is possible. It yields wave functions\nwhich are not, in general, orthogonal. However, closed-form results obtained\nafter Gram-Schmidt orthogonalization bear a striking resemblance to the exact\nanalytical expressions for the stationary-state wave functions of the various\npotential models considered (namely, a P\\\"{o}schl-Teller oscillator and the\nMorse oscillator).",
"arxiv_id": "quant-ph/0404102",
"authors": [
"Edward D. Davis"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.70.032101",
"journal_ref": "Phys. Rev. A 70, 032101 (2004)",
"title": "Sewing sound quantum flesh onto classical bones",
"url": "https://arxiv.org/abs/quant-ph/0404102"
},
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