additional symmetry operators A have connected the different ℓ multiplets among themselves, for a given energy (and C1), dictating states at each level. In effect, they have enlarged the angular momentum group SO(3) to SO(4)/Z2 ~ SO(3) × SO(3). Conservation and symmetry The conservation of the LRL vector corresponds to a subtle symmetry of the system. In classical mechanics, symmetries are continuous operations that map one orbit onto another without changing the energy of the system; in quantum mechanics, symmetries are continuous operations that "mix" electronic orbitals of the same energy, i.e., degenerate energy levels. A conserved quantity is usually associated with such symmetries. For example, every central force is symmetric under the rotation group SO(3), leading to the conservation of the angular momentum L. Classically, an overall rotation of the system does not affect the energy of an orbit; quantum mechanically, rotations mix the spherical harmonics of the same quantum number l without changing the energy. The symmetry for the inverse-square central force is higher and more subtle. The peculiar symmetry of the Kepler problem results in the conservation of both the angular momentum vector L and the LRL vector A (as defined above) and, quantum mechanically, ensures that the energy levels of hydrogen do not depend on the angular momentum quantum numbers l and m. The symmetry is more subtle, however, because the symmetry operation must take place in a higher-dimensional space; such symmetries are often called "hidden symmetries". Classically, the higher symmetry of the Kepler problem allows for continuous alterations of the orbits that preserve energy but not angular momentum; expressed another way, orbits of the same energy but different angular momentum (eccentricity) can be transformed continuously into one another. Quantum mechanically, this corresponds to mixing orbitals that differ in the l and m quantum numbers, such as the s () and p () atomic orbitals. Such mixing cannot be done with ordinary three-dimensional translations or rotations, but is equivalent to a rotation in a higher dimension. For negative energies – i.e., for bound systems – the higher symmetry group is SO(4), which preserves the length of four-dimensional vectors In 1935, Vladimir Fock showed that the quantum mechanical bound Kepler problem is equivalent to the problem of a free particle confined to a three-dimensional unit sphere in four-dimensional space. Specifically, Fock showed that the Schrödinger wavefunction in the momentum space for the Kepler problem was the stereographic