Angular Momentum Operator

: $\mathbf{L}=\mathbf{r} imes\mathbf{p}$ 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 : $\mathbf{L}=-i\hbar(\mathbf{r} imes abla)$ 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 : $= i \hbar \epsilon_{ijk} L_k$ , $\left L^2 ight = 0$ and, even more importantly, it Commutes with the Hamiltonian of such a chargeless and spinless particle : $\left H ight = 0$ . Angular momentum operators usually occur when solving a problem with Spherical Symmetry in Spherical Coordinates . Then, the angular momentum in space representation is: :: $\ L^2 = rac{1}{\sin heta}rac{\partial}{\partial heta}\left( \sin heta rac{\partial}{\partial heta} ight) + rac{1}{\sin^2 heta}rac{\partial^2}{\partial \phi^2}$ When solving to find Eigenstate s of this operator, we obtain the following
	:: <math> L Z L, M Ang	\hbar m l, m ang </math>
	::<math> \lang Heta , \phi L, M Ang	Y_{l,m}( heta,\phi)</math>

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