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This thesis is about the numerical approximation of the spectrum of the confined hydrogen atom. The energy levels and wave functions of the hydrogen atom are modelled by the corresponding eigenvalues and eigenvectors of the Schrödinger equation with Coulomb potential. At first different discretisation techniques for the hydrogenic Schrödinger equation are analysed. These include the finite difference method and the finite element method. Then appropriate methods for the solution of the discretised eigensystem are discussed. These include the Lanczos method, the implicitly restarted Arnoldi method and the Jacobi-Davidson method. A programme for the automatic computation of eigenspectra is implemented and numerical experiments are conducted to exemplify the theory. The programme is further on used to compute the spectrum of the hydrogen atom and several aspects of confinement are systematically investigated. These include the size and shape of the cavity as well as the position of the nucleus therein. It is shown that with increasing confinement all energy levels diverge towards positive infinity and that higher energy levels are affected first with decreasing domain size. Degeneracy is partially retained under confinement if the nucleus is positioned in the centre of the domain, but the spectrum becomes non-degenerate if the nucleus is shifted.