Energy levels of atomic hydrogen in a closed box: A natural cutoff criterion of the electronic partition function

The radial part of the Schr\"odinger equation for atomic hydrogen in a spherical box of radius $\ensuremath{\delta}$ is numerically solved. Two sets of energy levels are obtained, the first one reproduces the unperturbed (bound) levels up to a given principal quantum number while the other one (unbound) describes levels with energy greater than the unperturbed ionization energy of atomic hydrogen ${E}_{\text{H}}$. These last levels asymptotically converge to the corresponding set which can be obtained by the particle in the box model, i.e., levels which increase their energy as ${n}^{2}$ thus ensuring the convergence of the electronic partition function.

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