One important aspect of the study of Lie groups is their representations, that is, the way they can act (linearly) on vector spaces. In physics, Lie groups often encode the symmetries of a physical system. The way one makes use of this symmetry to help analyze the system is often through representation theory. Consider, for example, the time-independent Schrödinger equation in quantum mechanics, H ^ ^ ψ ψ = E ψ ψ . Assume the system in question has the rotation group SO(3) as a symmetry, meaning that the Hamiltonian operator H ^ ^ commutes with the action of SO(3) on the wave function ψ ψ . (One important example of such a system is the Hydrogen atom, which has a single spherical orbital.) This assumption does not necessarily mean that the solutions ψ ψ are rotationally invariant functions. Rather, it means that the space of solutions to H ^ ^ ψ ψ = E ψ ψ is invariant under rotations (for each fixed value of E ). This space, therefore, constitutes a representation of SO(3). These representations have been classified and the classification leads to a substantial simplification of the problem, essentially converting a three-dimensional partial differential equation to a one-dimensional ordinary differential equation.