Orbital-angular momentum


 * 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,

\boldsymbol{\ell}_k = \mathbf{r}_k\times \mathbf{p}_k , $$ where the symbol &times; indicates the cross product of two vectors. The total angular momentum of a system of N particles is

\mathbf{L} = \sum_{k=1}^N \boldsymbol{\ell}_k. $$ In the so-called x-representation of quantum mechanics, the vector rk is a multiplicative operator and

\mathbf{p}_k = - i \hbar \boldsymbol{\nabla}_k \equiv -i\hbar \left( \frac{\partial}{\partial x_k},\; \frac{\partial}{\partial y_k},\; \frac{\partial}{\partial z_k}\right), \qquad k=1,2, \ldots, N. $$ The components  of the orbital angular momentum satisfy the following commutation relations,

[L_x, \, L_y] = i\hbar L_z,\quad [L_z, \, L_x]= i\hbar L_y,\quad [L_y, \, L_z] = i\hbar L_z. $$ The fact that L is a conserved quantity is expressed by the commutation with the Hamiltonian (energy operator)

[H, L_\alpha] = 0, \quad \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.