Much insight in quantum mechanics can be gained from understanding the solutions to the time-dependent non-relativistic Schrödinger equation in an appropriate configuration space. In vector Cartesian coordinates
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or any equivalent formulation of this equation in a different coordinate system other than Cartesian coordinates. For example, systems with spherical symmetry are simplified when expressed with spherical coordinates. Very often, only numerical solutions to the Schrödinger equation can be found for a given physical system and its associated potential energy. Fortunately, there exists a subset of physical systems for which the form of the eigenfunctions and their associated energies can be found. These quantum-mechanical systems with analytical solutions are listed below.