Exactly Solvable Quantum Model with Spin-Dependent Coulomb Interaction
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918524201467904 |
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| author | Zhou, Jiang-Lin Zhang, Yu-Xuan Oh, Choo Hiap Chen, Jing-Ling |
| author_facet | Zhou, Jiang-Lin Zhang, Yu-Xuan Oh, Choo Hiap Chen, Jing-Ling |
| contents | In this work, we report an exactly solvable quantum model featuring a spin-dependent Coulomb interaction, described by the spin vector potential \(\vec{\mathcal{A}} = k (\vec{r} \times \vec{S}) / r^2\) together with a Coulomb-type scalar potential \(φ= κ/ r\). The model is governed by the Schrödinger-type Hamiltonian \(\mathcal{H}_{\rm S} = \vecΠ^2 / (2M) + q φ\) in nonrelativistic quantum mechanics and by the Dirac-type Hamiltonian \(\mathcal{H}_{\rm D} = c \vecα \cdot \vecΠ + βM c^2 + q φ\) in relativistic quantum mechanics, where \(\vecΠ = \vec{p} - (q/c)\vec{\mathcal{A}}\) is the canonical momentum. We demonstrate two main results: (i) Just as the Coulomb-type scalar potential \(\mathcal{S}_{\rm Maxwell} = \{\vec{\mathcal{A}} = 0,\ φ= κ/ r\}\) is an exact solution of Maxwell's equations, the gauge potential \(\mathcal{S}_{\rm YM} = \{\vec{\mathcal{A}} = k (\vec{r} \times \vec{S}) / r^2,\ φ= κ/ r\}\) constitutes an exact solution of the Yang--Mills equations. (ii) Both Hamiltonians \(\mathcal{H}_{\rm S}\) and \(\mathcal{H}_{\rm D}\) can be solved exactly in the presence of this spin-dependent Coulomb interaction. The resulting energy spectra are derived, and they naturally reduce to those of the ordinary hydrogen atom when the spin-dependent terms are neglected. Finally, we provide concluding remarks and discuss potential implications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_05103 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exactly Solvable Quantum Model with Spin-Dependent Coulomb Interaction Zhou, Jiang-Lin Zhang, Yu-Xuan Oh, Choo Hiap Chen, Jing-Ling Quantum Physics In this work, we report an exactly solvable quantum model featuring a spin-dependent Coulomb interaction, described by the spin vector potential \(\vec{\mathcal{A}} = k (\vec{r} \times \vec{S}) / r^2\) together with a Coulomb-type scalar potential \(φ= κ/ r\). The model is governed by the Schrödinger-type Hamiltonian \(\mathcal{H}_{\rm S} = \vecΠ^2 / (2M) + q φ\) in nonrelativistic quantum mechanics and by the Dirac-type Hamiltonian \(\mathcal{H}_{\rm D} = c \vecα \cdot \vecΠ + βM c^2 + q φ\) in relativistic quantum mechanics, where \(\vecΠ = \vec{p} - (q/c)\vec{\mathcal{A}}\) is the canonical momentum. We demonstrate two main results: (i) Just as the Coulomb-type scalar potential \(\mathcal{S}_{\rm Maxwell} = \{\vec{\mathcal{A}} = 0,\ φ= κ/ r\}\) is an exact solution of Maxwell's equations, the gauge potential \(\mathcal{S}_{\rm YM} = \{\vec{\mathcal{A}} = k (\vec{r} \times \vec{S}) / r^2,\ φ= κ/ r\}\) constitutes an exact solution of the Yang--Mills equations. (ii) Both Hamiltonians \(\mathcal{H}_{\rm S}\) and \(\mathcal{H}_{\rm D}\) can be solved exactly in the presence of this spin-dependent Coulomb interaction. The resulting energy spectra are derived, and they naturally reduce to those of the ordinary hydrogen atom when the spin-dependent terms are neglected. Finally, we provide concluding remarks and discuss potential implications. |
| title | Exactly Solvable Quantum Model with Spin-Dependent Coulomb Interaction |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2501.05103 |