|
| related topics |
| {equation, function, exp} |
| {cos, sin, state} |
| {group, space, representation} |
| {time, wave, function} |
| {theory, mechanics, state} |
| {energy, gaussian, time} |
| {states, state, optimal} |
| {classical, space, random} |
| {phase, path, phys} |
|
Sewing sound quantum flesh onto classical bones
Edward D. Davis
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).
- oai_identifier:
- oai:arXiv.org:quant-ph/0404102
- categories:
- quant-ph
- comments:
- RevTeX4, 6 pages
- doi:
- 10.1103/PhysRevA.70.032101
- arxiv_id:
- quant-ph/0404102
- journal_ref:
- Phys. Rev. A 70, 032101 (2004)
- created:
- 2004-04-19
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