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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2211.16149 |
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| _version_ | 1866913471548882944 |
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| author | Berger, Raphael J. F. |
| author_facet | Berger, Raphael J. F. |
| contents | The problem of a spin-free electron with mass $m$, charge $e$ confined onto a ring of radius $R_0$ and with an attractive Dirac delta potential with scaling factor (depth) $κ$ in non-relativistic theory has closed form analytical solutions. The single bound state function is of the form of a hyperbolic cosine that however contains a parameter $d>0$ which is the single positive real solution of the transcendental equation $\coth(d) = λd$ for non zero real $λ=\frac{2}{πκ}$. The energy eigenvalue of the bound state $\varepsilon=-\frac{d^2}{2π^2}\approx \frac{q e m R_0}{2 \hbar^2}$. In addition a discretly infinite set of unbounded solutions exists, formally these solutions are obtained from the terms for the bound solution by substituting $d \to i d $ yielding $\cot(d) = λd$ as characteristic equation with the corresponding set of solutions $d_k, k\in\mathbb{N}$, the respective state functions can be obtained via $\cosh(x)\overset{x \to i x}{\longrightarrow}\cos(x)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_16149 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Quantum Mechanical Problem of a Particle on a Ring with Delta Well Berger, Raphael J. F. Quantum Physics The problem of a spin-free electron with mass $m$, charge $e$ confined onto a ring of radius $R_0$ and with an attractive Dirac delta potential with scaling factor (depth) $κ$ in non-relativistic theory has closed form analytical solutions. The single bound state function is of the form of a hyperbolic cosine that however contains a parameter $d>0$ which is the single positive real solution of the transcendental equation $\coth(d) = λd$ for non zero real $λ=\frac{2}{πκ}$. The energy eigenvalue of the bound state $\varepsilon=-\frac{d^2}{2π^2}\approx \frac{q e m R_0}{2 \hbar^2}$. In addition a discretly infinite set of unbounded solutions exists, formally these solutions are obtained from the terms for the bound solution by substituting $d \to i d $ yielding $\cot(d) = λd$ as characteristic equation with the corresponding set of solutions $d_k, k\in\mathbb{N}$, the respective state functions can be obtained via $\cosh(x)\overset{x \to i x}{\longrightarrow}\cos(x)$. |
| title | The Quantum Mechanical Problem of a Particle on a Ring with Delta Well |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2211.16149 |