Eigenvalues and eigenvectors for a hermitian gaussian operator: Role of the Schrödinger-Robertson uncertainty relation
The eigenvalues and eigenvectors of a normalized gaussian operator do not seem to have been previously considered. I solve this problem for 1-dimensional translational systems. I also address the question as to whether a gaussian operator is a density operator. To answer that question, it is first n...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-08-01
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| Series: | Electronic Research Archive |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/era.2023281?viewType=HTML |
| _version_ | 1797657973521121280 |
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| author | R. F. Snider |
| author_facet | R. F. Snider |
| author_sort | R. F. Snider |
| collection | DOAJ |
| description | The eigenvalues and eigenvectors of a normalized gaussian operator do not seem to have been previously considered. I solve this problem for 1-dimensional translational systems. I also address the question as to whether a gaussian operator is a density operator. To answer that question, it is first necessary to be sure what conditions must be satisfied, so a short review of density operators is given. Since position and momentum do not commute in quantum mechanics, it is useful to start with the consequences of the noncommutation, which is generally the Schrödinger-Robertson uncertainty relation, which is also briefly reviewed. It is found that the question of whether a gaussian operator is a density operator is directly tied to this uncertainty relation. Since the Wigner function is the phase space representation of a translational density operator, it is natural to consider the gaussian phase space function associated with a gaussian operator and to compare the phase space and operator properties. Throughout such discussions, the independent parameters in these functions are the first and second moments of position and momentum. The application of this formalism to the free translation and spreading of a gaussian packet is given and shows the formal similarity between classical and quantum behavior, whereas the literature standardly only considers the pure state case (equivalent to a single wavefunction). |
| first_indexed | 2024-03-11T17:52:19Z |
| format | Article |
| id | doaj.art-c1f4fef22199441ba89af4e0d6e858e0 |
| institution | Directory Open Access Journal |
| issn | 2688-1594 |
| language | English |
| last_indexed | 2024-03-11T17:52:19Z |
| publishDate | 2023-08-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Electronic Research Archive |
| spelling | doaj.art-c1f4fef22199441ba89af4e0d6e858e02023-10-18T01:46:54ZengAIMS PressElectronic Research Archive2688-15942023-08-013195541555810.3934/era.2023281Eigenvalues and eigenvectors for a hermitian gaussian operator: Role of the Schrödinger-Robertson uncertainty relationR. F. Snider0Department of Chemitry, University of British Columbia, Vancouver, BC, V6T 1Z1, CanadaThe eigenvalues and eigenvectors of a normalized gaussian operator do not seem to have been previously considered. I solve this problem for 1-dimensional translational systems. I also address the question as to whether a gaussian operator is a density operator. To answer that question, it is first necessary to be sure what conditions must be satisfied, so a short review of density operators is given. Since position and momentum do not commute in quantum mechanics, it is useful to start with the consequences of the noncommutation, which is generally the Schrödinger-Robertson uncertainty relation, which is also briefly reviewed. It is found that the question of whether a gaussian operator is a density operator is directly tied to this uncertainty relation. Since the Wigner function is the phase space representation of a translational density operator, it is natural to consider the gaussian phase space function associated with a gaussian operator and to compare the phase space and operator properties. Throughout such discussions, the independent parameters in these functions are the first and second moments of position and momentum. The application of this formalism to the free translation and spreading of a gaussian packet is given and shows the formal similarity between classical and quantum behavior, whereas the literature standardly only considers the pure state case (equivalent to a single wavefunction).https://www.aimspress.com/article/doi/10.3934/era.2023281?viewType=HTMLeigenvalueeigenvectorgaussiandensity operator |
| spellingShingle | R. F. Snider Eigenvalues and eigenvectors for a hermitian gaussian operator: Role of the Schrödinger-Robertson uncertainty relation Electronic Research Archive eigenvalue eigenvector gaussian density operator |
| title | Eigenvalues and eigenvectors for a hermitian gaussian operator: Role of the Schrödinger-Robertson uncertainty relation |
| title_full | Eigenvalues and eigenvectors for a hermitian gaussian operator: Role of the Schrödinger-Robertson uncertainty relation |
| title_fullStr | Eigenvalues and eigenvectors for a hermitian gaussian operator: Role of the Schrödinger-Robertson uncertainty relation |
| title_full_unstemmed | Eigenvalues and eigenvectors for a hermitian gaussian operator: Role of the Schrödinger-Robertson uncertainty relation |
| title_short | Eigenvalues and eigenvectors for a hermitian gaussian operator: Role of the Schrödinger-Robertson uncertainty relation |
| title_sort | eigenvalues and eigenvectors for a hermitian gaussian operator role of the schrodinger robertson uncertainty relation |
| topic | eigenvalue eigenvector gaussian density operator |
| url | https://www.aimspress.com/article/doi/10.3934/era.2023281?viewType=HTML |
| work_keys_str_mv | AT rfsnider eigenvaluesandeigenvectorsforahermitiangaussianoperatorroleoftheschrodingerrobertsonuncertaintyrelation |