Orbital-angular momentum

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See also angular momentum in quantum mechanics

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 rk = (xk, yk, zk) and if the momentum of the same particle is pk, then the orbital angular momentum of particle k is defined as the following vector operator,

ℓk=𝐫k×𝐩k,

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

𝐋=∑k=1Nℓk.

In the so-called x-representation of quantum mechanics, the vector rk is a multiplicative operator and

𝐩k=−iℏ∇k≡−iℏ(∂∂xk,∂∂yk,∂∂zk),k=1,2,…,N.

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

[Lx,Ly]=iℏLz,[Lz,Lx]=iℏLy,[Ly,Lz]=iℏLz.

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

[H,Lα]=0,α=x,y,z.

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