Angular momentum is an important concept in both classical and quantum mechanics. Its value is preserved throughout the motion of most relevant systems.The angular momentum L of a particle of mass m with linear momentum p with a radial vector r with respect to a center of rotation is defined as

L = r×p

where the display of a symbol in red denotes that it is a vector. The above vector product can be represented in the form of a determinant; i.e.,

| ux	uy	uz |
| rx	ry	rz |
| px	py	px |

where u_x, u_y and u_z are unit vectors in the directions of the x, y and z axes, respectively.

For further developments it greatly simplifies things if the components of the vector r are represented as (x, y, z). The components of L are then given by

L_x = yp_z − zp_y
L_y = zp_x − xp_z
L_z = xp_y − yp_x

The operators associated with these components are generated by replacing a momentum component p_w with −ih(∂/∂w), where w can be x, y or z and h is Planck's constant divided by 2π. Thus

L_x = −ih[y(∂/∂z) − z(∂/∂y)]
L_y = −ih[z(∂/∂x) − x(∂/∂z)]
L_z = −ih[x(∂/∂y) − y(∂/∂x)]

Before proceding further it is necessary to cover the algebra of operators.

Operator Algebra

An operator is a mapping that takes a function as an input and gives a function as an output. Partial derivatives are operators and so is the operation of multiplying a function by a specific function.

The commutator of two operators P and Q is denoted as [ P, Q] and is given by

[P, Q]ψ = P(Qψ) − Q(Pψ)

for any function ψ.

Some of the elementary properties of the commutator are

• The anticommutative property:

  [Q, P] = −[P, Q]
  and hence
  [P, P] = 0

• The distributive property

  [P, Q+R] = [P, Q] + [P, R]
  and
  [P+Q, R] = [P, R] + [Q, R]

• The product of operators property

  [P, QR] = [P, Q]R + Q[P, R]

Consider now [w, ∂/∂u] where w and u can be any of the coordinates x, y or z. Then

[w, ∂/∂u]ψ = w(∂ψ/∂u) − ∂(wψ)/∂u
= w(∂ψ/∂u) − w(∂ψ/∂u) − (∂w/∂u)ψ
which reduces to
[w, ∂/∂u]ψ = −δ_w,uψ
where δ_w,u=1 if w=u
and 0 otherwise

Momentum Operators in Quantum Mechanics

In quantum mechanics a momentum component p_u is replaced by −ih(∂/∂u). Thus for any u and w in the set {x, y, z}

[w, p_u]ψ = wp_uψ − p_u(wψ)
= −ihw(∂ψ/∂u) − [−ih(∂(wψ)/∂u)]
= −ih[w(∂ψ/∂u) + (∂(wψ)/∂u)]
= ih[w, (∂/∂u)]ψ
= − ihδ_{w, u}ψ

These operator relations may now be applied to evaluating the commutators of the components of angular momentum. Consider first [L_x, L_y].

[L_x, L_y] = [yp_z−zp_y, zp_x−xp_z]
and with repeted applications
of the distributive property rule
= [yp_z, zp_x] − [yp_z, xp_z] − [zp_y, zp_x] + [zp_y, xp_z]

With an application of the product property and the result that [P, P]=0 the term [yp_z, xp_z] evaluates to the zero operator. Likewise for the term [zp_y, zp_x]. Thus

[L_x, L_y] = [yp_z, zp_x] + [zp_y, xp_z]
and with the application of
the product property rule
= y[p_z, z]p_x + p_y[z, p_z]x
or, applying the anticommutative
property to [p_z, z]
= p_y[z, p_z]x − y[z, p_z, z]p_x
and with the replacement
of [z, p_z] with ih
= ih(xp_y − yp_y)
= ihL_z

With a cyclic permutation of the indices the commutators of the three components of angular momentum are found to be

[L_x, L_y] = = ihL_z
[L_y, L_z] = = ihL_x
[L_z, L_x] = = ihL_y

The Commutivity of Angular Momentum Operators

Two operators, P and Q, commute; i.e., PQ=QP, if [P, Q]=0.

Let

L² = L_x² + L_y² + L_z²

Now consider

[L_z, L²] = [L_z, L_x²] + [L_z, L_y²] + [L_z, L_z²]

The last term, involving repeated applications of L_z reduces to the zero operator. Thus

[L_z, L²] = [L_z, L_x²] + [L_z, L_y²]
and with application of the product property
[L_z, L²] = L_x[L_z, L_z] + [L_z, L_x]L_x + L_y[L_z, L_y] + [L_z, L_y]L_y
then replacing the commutators with
the relations previously developed
[L_z, L²] = ih([L_xL_y + L_yL_x − L_yL_x − L_xL_y) = 0

Likewise [L_x, L²]=0 and [L_y, L²]=0.

Although it is not proven here, the condition [L_z, L²]=0 means that L_z and L² the same eigenfunctions. The eigenvalues of L² are of the form hl(l+1) for l an integer. The eigenvalues of L_z are of the form hm where m is an integer −l≤m≤l.