Angular momentum operator

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In quantum mechanics, the angular momentum operator is an operator that is the quantum analog of the classical angular momentum. It plays a central role in the theory of atomic physics and other quantum problems with rotational symmetry.

1 Definition
2 In spherical coordinates
3 In classical physics
4 See also

In quantum mechanics, the angular momentum is defined like momentum - not as a quantity but as an operator on the wave function:

$\mathbf{L}=\mathbf{r}\times\mathbf{p}$

where r and p are the position and momentum operators respectively. In particular, for a single particle with no electric charge and no spin, the angular momentum operator can be written in the position basis as

$\mathbf{L}=-i\hbar(\mathbf{r}\times\nabla)$

where is the gradient operator. This is a commonly encountered form of the angular momentum operator, though not the most general one. It has the following properties

$[L_i, L_j ] = i \hbar \epsilon_{ijk} L_k$

$\left[L_i, L^2 \right] = 0$

and, even more importantly, it commutes with the Hamiltonian of such a chargeless and spinless particle

$\left[L_i, H \right] = 0$ .

The first commutation relation is an example of what is generally known as a Lie algebra. In this case, the Lie algebra is that of SU(2) or SO(3), the rotation group in three dimensions. The second commutation relation indicates that $L 2$ is a Casimir invariant. The third commutation relation states that the angular momentum is a constant of motion, and is a special case of Liouville's equation for quantum mechanics, or more precisely, of Ehrenfest's theorem.

Angular momentum operators usually occur when solving a problem with spherical symmetry in spherical coordinates. Then, the angular momentum in space representation is:

$\ \frac{1}{-\hbar^2}L^2 = \frac{1}{\sin\theta}\frac{\partial}{\partial \theta}\left( \sin\theta \frac{\partial}{\partial \theta}\right) + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial \phi^2}$