Angular momentum operator

In quantum mechanics, the angular momentum operator is an operator analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic physics and other quantum problems involving rotational 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.

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 a wave function, rather than on a point or rigid body as classical calculations entail. This is because at the scale of quantum mechanics, the matter 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 the cross product of a wave function's position operator (r) and momentum operator (p):

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

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:

$mathbf\{L\}=-ihbar(mathbf\{r\}timesnabla)$

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 and tensor calculus.

When using Cartesian 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:

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

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:

$left[L\_i,\; H\; right]\; =\; 0$

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 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.

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 and Dirac notation.

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

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

When solving to find eigenstates of this operator, we obtain the following

$L^2\; |\; l,\; m\; rang\; =\; \{hbar\}^2\; l(l+1)\; |\; l,\; m\; rang$

$L\_z\; |\; l,\; m\; rang\; =\; hbar\; m\; |\; l,\; m\; rang$

where

$lang\; theta\; ,\; phi\; |\; l,\; m\; rang\; =\; Y\_\{l,m\}(theta,phi)$

are the spherical harmonics.

See 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