| dbp:proof
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- A symmetric tensor operator analogous to the Laplace–Runge–Lenz vector for the Kepler problem may be defined,
:
which commutes with the Hamiltonian,
:
Since it commutes with the Hamiltonian , it represents 6−1=5 constants of motion.
It has the following properties,
:
:
:
Apart from the tensorial trace of the operator, which is the Hamiltonian, the remaining 5 operators can be rearranged into their spherical component form as
:
:
:
Further, the angular momentum operators are written in spherical component form as
:
:
They obey the following commutation relations,
:
:
:
:
:
:
The eight operators obey the same commutation relations as the infinitesimal generators of the group, detailed above.
As such, the symmetry group of Hamiltonian for a linear isotropic 3D Harmonic oscillator is isomorphic to group. (en)
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