We often use the Cartesian coordinate system, since it is the naive approach at macro level (placing a box just "next to" or "above" the other box). There are, however, many more such systems, incl. these here: http://en.wikipedia.org/wiki/Coordinate_system.
Which of the systems allow modeling experimental data in the most effective way (exploiting the most explicit characteristics of quantum-level interactions)?
Answer
In general, in physics the question which coordinate system is best suited to approach your problem depends mostly on the symmetry of your problem. If, for example, there is a complete (spatial) rotational symmetry, the obvious choice for where some sort of simplification might be expected is spherical coordinates. The same holds in quantum mechanics: The simple model of the hydrogen atom is a perfect example of a problem that is simpler in spherical coordinates.
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