On the Stochastic Mechanics Foundation of Quantum Mechanics
Among the famous formulations of quantum mechanics, the stochastic picture developed since the middle of the last century remains one of the less known ones. It is possible to describe quantum mechanical systems with kinetic equations of motion in configuration space based on conservative diffusion...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-05-01
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| Series: | Universe |
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| Online Access: | https://www.mdpi.com/2218-1997/7/6/166 |
| _version_ | 1797532311764336640 |
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| author | Michael Beyer Wolfgang Paul |
| author_facet | Michael Beyer Wolfgang Paul |
| author_sort | Michael Beyer |
| collection | DOAJ |
| description | Among the famous formulations of quantum mechanics, the stochastic picture developed since the middle of the last century remains one of the less known ones. It is possible to describe quantum mechanical systems with kinetic equations of motion in configuration space based on conservative diffusion processes. This leads to the representation of physical observables through stochastic processes instead of self-adjoint operators. The mathematical foundations of this approach were laid by Edward Nelson in 1966. It allows a different perspective on quantum phenomena without necessarily using the wave-function. This article recaps the development of stochastic mechanics with a focus on variational and extremal principles. Furthermore, based on recent developments of optimal control theory, the derivation of generalized canonical equations of motion for quantum systems within the stochastic picture are discussed. These so-called quantum Hamilton equations add another layer to the different formalisms from classical mechanics that find their counterpart in quantum mechanics. |
| first_indexed | 2024-03-10T10:57:20Z |
| format | Article |
| id | doaj.art-38df03e018134d969bf5a6d4d749b5bb |
| institution | Directory Open Access Journal |
| issn | 2218-1997 |
| language | English |
| last_indexed | 2024-03-10T10:57:20Z |
| publishDate | 2021-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Universe |
| spelling | doaj.art-38df03e018134d969bf5a6d4d749b5bb2023-11-21T21:43:48ZengMDPI AGUniverse2218-19972021-05-017616610.3390/universe7060166On the Stochastic Mechanics Foundation of Quantum MechanicsMichael Beyer0Wolfgang Paul1Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle (Saale), GermanyInstitut für Physik, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle (Saale), GermanyAmong the famous formulations of quantum mechanics, the stochastic picture developed since the middle of the last century remains one of the less known ones. It is possible to describe quantum mechanical systems with kinetic equations of motion in configuration space based on conservative diffusion processes. This leads to the representation of physical observables through stochastic processes instead of self-adjoint operators. The mathematical foundations of this approach were laid by Edward Nelson in 1966. It allows a different perspective on quantum phenomena without necessarily using the wave-function. This article recaps the development of stochastic mechanics with a focus on variational and extremal principles. Furthermore, based on recent developments of optimal control theory, the derivation of generalized canonical equations of motion for quantum systems within the stochastic picture are discussed. These so-called quantum Hamilton equations add another layer to the different formalisms from classical mechanics that find their counterpart in quantum mechanics.https://www.mdpi.com/2218-1997/7/6/166stochastic mechanicsquantum mechanicsstochastic foundation of quantum mechanicsstochastic differential equations |
| spellingShingle | Michael Beyer Wolfgang Paul On the Stochastic Mechanics Foundation of Quantum Mechanics Universe stochastic mechanics quantum mechanics stochastic foundation of quantum mechanics stochastic differential equations |
| title | On the Stochastic Mechanics Foundation of Quantum Mechanics |
| title_full | On the Stochastic Mechanics Foundation of Quantum Mechanics |
| title_fullStr | On the Stochastic Mechanics Foundation of Quantum Mechanics |
| title_full_unstemmed | On the Stochastic Mechanics Foundation of Quantum Mechanics |
| title_short | On the Stochastic Mechanics Foundation of Quantum Mechanics |
| title_sort | on the stochastic mechanics foundation of quantum mechanics |
| topic | stochastic mechanics quantum mechanics stochastic foundation of quantum mechanics stochastic differential equations |
| url | https://www.mdpi.com/2218-1997/7/6/166 |
| work_keys_str_mv | AT michaelbeyer onthestochasticmechanicsfoundationofquantummechanics AT wolfgangpaul onthestochasticmechanicsfoundationofquantummechanics |