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 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,

${\displaystyle {\mathit{\ell }}_k={\mathbf{𝐫}}_k\times {\mathbf{𝐩}}_k,}$

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

${\displaystyle \mathbf{𝐋}={\displaystyle \sum _{k=1}^N}{\mathit{\ell }}_k.}$

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

${\displaystyle {\mathbf{𝐩}}_k=-i\hslash {\mathrm{\nabla }}_k\equiv -i\hslash (\frac{\partial }{\partial x_k},\frac{\partial }{\partial y_k},\frac{\partial }{\partial z_k}),k=1,2,\dots ,N.}$

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

${\displaystyle [L_x,L_y]=i\hslash L_z,[L_z,L_x]=i\hslash L_y,[L_y,L_z]=i\hslash L_z.}$

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

${\displaystyle [H,L_{\alpha }]=0,\alpha =x,y,z.}$

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