Electromagnetic Transitions in the Spectrum of a Confined Hydrogen Atom

Transition probabilities per time unit in the spectrum of the hydrogen atom confined in a spherical cavity are studied with the Dirac equation. The calculations are performed with simple and accurate expansions of the Dirac components provided by the Lagrange-mesh method with small numbers of scaled Jacobi mesh points. The initial and final states as well as intermediate states occurring in two-photon transitions are described with a single Lagrange mesh. With this simplification, accurate matrix elements are easily computed with the Gauss quadrature associated with the Lagrange–Jacobi mesh. One-photon transitions are considered from the n=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n = 2$$\end{document} and 3 levels. Most transition probabilities increase as a negative power of the confinement radius when it becomes small. The two-photon 2s1/2-1s1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2s_{1/2}-1s_{1/2}$$\end{document} transition stops being the main deexcitation channel of the 2s1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2s_{1/2}$$\end{document} level when the confinement radius decreases below about 15 atomic units. The 2p1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2p_{1/2}$$\end{document} and 2p3/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2p_{3/2}$$\end{document} levels come below the 2s1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2s_{1/2}$$\end{document} level and allow a faster one-photon deexcitation.