# Angular momentum operator

In

quantum mechanics , the**angular momentum operator**is anoperator analogous to classicalangular momentum . The angular momentum operator plays a central role in the theory ofatomic physics and other quantum problems involvingrotational symmetry . In both classical and quantum mechanical systems, angular momentum (together with linear momentum and energy) is one of the three fundamental properties of motion.Introductory Quantum Mechanics, Richard L. Liboff, 2nd Edition, ISBN 0201547155]**Intuitive meaning**Angular momentum quantifies the rotational aspect of motion. Like energy and linear momentum, angular momentum in an isolated system is conserved. The concept of an angular momentum

operator is necessary in quantum mechanics, as calculations of angular momentum must be made upon awave function , rather than on a point or rigid body as classical calculations entail. This is because at the scale of quantum mechanics, thematter analyzed is best described by a wave equation or probability amplitude, rather than as a collection of fixed points or as a rigid body.Vector calculus is used in calculations of angular momentum, as angular momentum has compenents in each of the three spatial dimensions.**Mathematical definition**Angular momentum

**L**is mathematically defined as thecross product of a wave function's position operator (**r**) and momentum operator (**p**)::$mathbf\{L\}=mathbf\{r\}\; imesmathbf\{p\}$

In the special case of a single particle with no

electric charge and no spin, the angular momentum operator can be written in the position basis as a single vector equation::$mathbf\{L\}=-ihbar(mathbf\{r\}\; imes\; abla)$

where is the

gradient operator. This is a commonly encountered form of the angular momentum operator, though not the most general one.**Commutator relations of Cartesian components**:"This section includes mathematical equations involving

vector calculus andtensor calculus ."When usingCartesian coordinates , it is customary to refer to the three spatial components of the angular momentum operator as $L\_i$, $L\_j$ and $L\_k$. The angular momentum operator has the following commutation properties with respect to its individual components::$[L\_i,\; L\_j\; ]\; =\; i\; hbar\; epsilon\_\{ijk\}\; L\_k$

where $epsilon\_\{ijk\}$ denotes the

Levi-Civita symbol .

However, the square of the total angular momentum ($L^2$) (defined as the sum of the squares of the three Cartesian components) commutes with its components as follows: :$left\; [L\_i,\; L^2\; ight]\; =\; 0$This means that no two individual components of quantum angular momentum can be simultaneously specified for a given system, whereas the total angular momentum can be simultaneously specified along with any

**one**of the operator's components. The lack of commutation of the individual components of the angular momentum describe what is known in physics as an uncertainty principle.

Even more importantly, the angular momentum operator commutes with the Hamiltonian of such a chargeless and spinless particle::$left\; [L\_i,\; H\; ight]\; =\; 0$

The Hamiltonian "H" represents the energy of the system and is used to generate translations through time. Thus, operators which commute with "H" represent conserved quantities.

**Further analysis of commutation properties**The first commutation relation above is an example of what is generally known as a

Lie algebra . In this case, the Lie algebra is that ofSU(2) orSO(3) , therotation group in three dimensions. The second commutation relation indicates that $L^2$ is aCasimir invariant . The third commutation relation states that the angular momentum is aconstant of motion , and is a special case of Liouville's equation forquantum mechanics , or more precisely, ofEhrenfest's theorem .**In classical physics**It should be noted that the angular momentum in

classical mechanics obeys a similar commutation relation,:$\{L\_i,\; L\_j\; \}\; =\; epsilon\_\{ijk\}\; L\_k\; !$

where $\{\; ,\}$ is the

Poisson bracket .**Angular momentum computations in spherical coordinates**:"This section includes mathematical equations involving

partial differential equations andDirac notation ."Angular momentum operators usually occur when solving a problem withspherical symmetry inspherical coordinates . Then, the angular momentum in space representation is::: $frac\{1\}\{-hbar^2\}L^2\; =\; frac\{1\}\{sin\; heta\}frac\{partial\}\{partial\; heta\}left(\; sin\; heta\; frac\{partial\}\{partial\; heta\}\; ight)\; +\; frac\{1\}\{sin^2\; heta\}frac\{partial^2\}\{partial\; phi^2\}$When solving to findeigenstate s of this operator, we obtain the following:: $L^2\; |\; l,\; m\; ang\; =\; \{hbar\}^2\; l(l+1)\; |\; l,\; m\; ang$:: $L\_z\; |\; l,\; m\; ang\; =\; hbar\; m\; |\; l,\; m\; ang$where::$lang\; heta\; ,\; phi\; |\; l,\; m\; ang\; =\; Y\_\{l,m\}(\; heta,phi)$are thespherical harmonic s.**ee also***

Runge-Lenz vector (used to describe the shape and orientation of bodies in orbit)

*Position operator

*Momentum operator

*Annihilation operator

*Creation operator

*Hamiltonian operator

*Ladder operator **References**

*Wikimedia Foundation.
2010.*

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