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# Orbital-angular momentum

In quantum mechanics, **orbital angular momentum** is a conserved property of a system of one or more particles that are in a centrally symmetric potential. If the radius of particle *k* with respect to the center of symmetry is **r**_{k} = (*x*_{k}, *y*_{k}, *z*_{k}) and if the momentum of the same particle is **p**_{k}, then the orbital angular momentum of particle *k* is defined as the following vector operator,

where the symbol × indicates the cross product of two vectors.
The total angular momentum of a system of *N* particles is

In the so-called *x*-representation of quantum mechanics, the vector **r**_{k} is a multiplicative operator and

The components of the orbital angular momentum satisfy the following commutation relations,

The fact that **L** is a conserved quantity is expressed by the commutation with the Hamiltonian (energy operator)

It can be shown that this condition is necessary and sufficient that the potential energy part of *H* be centrally symmetric.