Quantum Mechanics and Liouville's Equation
In non-relativistic quantum mechanics, the absolute square of Schrödinger's wave function for a particle in a potential determines the probability of finding it either at a position or momentum at a given time. In classical mechanics the corresponding problem is determined by the solution of Li...
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| Format: | Article |
| Language: | English |
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Quanta
2017-09-01
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| Series: | Quanta |
| Online Access: | http://quanta.ws/ojs/index.php/quanta/article/view/63 |
| _version_ | 1818939177002074112 |
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| author | Michael Nauenberg |
| author_facet | Michael Nauenberg |
| author_sort | Michael Nauenberg |
| collection | DOAJ |
| description | In non-relativistic quantum mechanics, the absolute square of Schrödinger's wave function for a particle in a potential determines the probability of finding it either at a position or momentum at a given time. In classical mechanics the corresponding problem is determined by the solution of Liouville's equation for the probability density of finding the joint position and momentum of the particle at a given time. Integrating this classical solution over either one of these two variables can then be compared with the probability in quantum mechanics. For the special case that the force is a constant, it is shown analytically that for an initial Gaussian probability distribution, the solution of Liouville's integrated over momentum is equal to Schrödinger's probability function in coordinate space, provided the coordinate and momentum initial widths of this classical solution satisfy the minimal Heisenberg uncertainty relation. Likewise, integrating Lioville's solution over position is equal to Schrödinger's probability function in momentum space.
Quanta 2017; 6: 53–56. |
| first_indexed | 2024-12-20T06:19:36Z |
| format | Article |
| id | doaj.art-bfde47fa7f1d420c8091c1c1ba73cd72 |
| institution | Directory Open Access Journal |
| issn | 1314-7374 |
| language | English |
| last_indexed | 2024-12-20T06:19:36Z |
| publishDate | 2017-09-01 |
| publisher | Quanta |
| record_format | Article |
| series | Quanta |
| spelling | doaj.art-bfde47fa7f1d420c8091c1c1ba73cd722022-12-21T19:50:27ZengQuantaQuanta1314-73742017-09-0161535610.12743/quanta.v6i1.6333Quantum Mechanics and Liouville's EquationMichael Nauenberg0University of California, Santa CruzIn non-relativistic quantum mechanics, the absolute square of Schrödinger's wave function for a particle in a potential determines the probability of finding it either at a position or momentum at a given time. In classical mechanics the corresponding problem is determined by the solution of Liouville's equation for the probability density of finding the joint position and momentum of the particle at a given time. Integrating this classical solution over either one of these two variables can then be compared with the probability in quantum mechanics. For the special case that the force is a constant, it is shown analytically that for an initial Gaussian probability distribution, the solution of Liouville's integrated over momentum is equal to Schrödinger's probability function in coordinate space, provided the coordinate and momentum initial widths of this classical solution satisfy the minimal Heisenberg uncertainty relation. Likewise, integrating Lioville's solution over position is equal to Schrödinger's probability function in momentum space. Quanta 2017; 6: 53–56.http://quanta.ws/ojs/index.php/quanta/article/view/63 |
| spellingShingle | Michael Nauenberg Quantum Mechanics and Liouville's Equation Quanta |
| title | Quantum Mechanics and Liouville's Equation |
| title_full | Quantum Mechanics and Liouville's Equation |
| title_fullStr | Quantum Mechanics and Liouville's Equation |
| title_full_unstemmed | Quantum Mechanics and Liouville's Equation |
| title_short | Quantum Mechanics and Liouville's Equation |
| title_sort | quantum mechanics and liouville s equation |
| url | http://quanta.ws/ojs/index.php/quanta/article/view/63 |
| work_keys_str_mv | AT michaelnauenberg quantummechanicsandliouvillesequation |