dorsal/arxiv
View SchemaGeneric Bohmian Trajectories of an Isolated Particle
| Authors | D. M. Appleby |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9905003 |
| URL | https://arxiv.org/abs/quant-ph/9905003 |
| DOI | 10.1023/A:1018842401049 |
| Journal | Found.Phys.29:1863-1916,1999 |
Abstract
The generic Bohmian trajectories are calculated for an isolated particle in an approximate energy eigenstate, for an arbitrary one-dimensional potential well. It is shown, that the necessary and sufficient condition for there to be a negligible probability of the trajectory deviating significantly from the classical trajectory at any stage in the motion is, that the state be a narrowly localised wave packet. The properties of the Bohmian trajectories are discussed in relationship to the theory of retrodictively optimal simultaneous measurements of position and momentum which was presented in several previous papers. It is shown that the Bohmian velocity at x is the expectation value of the velocity which would be observed at x, if one were to make a retrodictively optimal simultaneous measurement of x and p, in the limit as the error in the measurement of x tends to zero. This explains the tendency of the Bohmian particle to behave in a highly non-classical manner. It also explains why the trajectories in the interpretation recently proposed by Garcia de Polavieja tend to be much more nearly classical in the limit of large quantum number. The implications for other trajectory interpretations are considered.
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"abstract": "The generic Bohmian trajectories are calculated for an isolated particle in\nan approximate energy eigenstate, for an arbitrary one-dimensional potential\nwell. It is shown, that the necessary and sufficient condition for there to be\na negligible probability of the trajectory deviating significantly from the\nclassical trajectory at any stage in the motion is, that the state be a\nnarrowly localised wave packet. The properties of the Bohmian trajectories are\ndiscussed in relationship to the theory of retrodictively optimal simultaneous\nmeasurements of position and momentum which was presented in several previous\npapers. It is shown that the Bohmian velocity at x is the expectation value of\nthe velocity which would be observed at x, if one were to make a retrodictively\noptimal simultaneous measurement of x and p, in the limit as the error in the\nmeasurement of x tends to zero. This explains the tendency of the Bohmian\nparticle to behave in a highly non-classical manner. It also explains why the\ntrajectories in the interpretation recently proposed by Garcia de Polavieja\ntend to be much more nearly classical in the limit of large quantum number. The\nimplications for other trajectory interpretations are considered.",
"arxiv_id": "quant-ph/9905003",
"authors": [
"D. M. Appleby"
],
"categories": [
"quant-ph"
],
"doi": "10.1023/A:1018842401049",
"journal_ref": "Found.Phys.29:1863-1916,1999",
"title": "Generic Bohmian Trajectories of an Isolated Particle",
"url": "https://arxiv.org/abs/quant-ph/9905003"
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