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Angular Momentum in Quantum Mechanics

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.,

| uxuyuz |
| rxryrz |
| pxpypx |

where ux, uy and uz 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

Lx = ypz − zpy
Ly = zpx − xpz
Lz = xpy − ypx

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

Lx = −ih[y(∂/∂z) − z(∂/∂y)]
Ly = −ih[z(∂/∂x) − x(∂/∂z)]
Lz = −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

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 = −δw,uψ
where δw,u=1 if w=u
and 0 otherwise

Momentum Operators in Quantum Mechanics

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

[w, pu]ψ = wpuψ − pu(wψ)
= −ihw(∂ψ/∂u) − [−ih(∂(wψ)/∂u)]
= −ih[w(∂ψ/∂u) + (∂(wψ)/∂u)]
= ih[w, (∂/∂u)]ψ
= − ihδw, uψ

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

[Lx, Ly] = [ypz−zpy, zpx−xpz]
and with repeted applications
of the distributive property rule
= [ypz, zpx] − [ypz, xpz] − [zpy, zpx] + [zpy, xpz]

With an application of the product property and the result that [P, P]=0 the term [ypz, xpz] evaluates to the zero operator. Likewise for the term [zpy, zpx]. Thus

[Lx, Ly] = [ypz, zpx] + [zpy, xpz]
and with the application of
the product property rule
= y[pz, z]px + py[z, pz]x
or, applying the anticommutative
property to [pz, z]
= py[z, pz]x − y[z, pz, z]px
and with the replacement
of [z, pz] with ih
= ih(xpy − ypy)
= ihLz

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

[Lx, Ly] = = ihLz
[Ly, Lz] = = ihLx
[Lz, Lx] = = ihLy

The Commutivity of Angular Momentum Operators

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


L² = Lx² + Ly² + Lz²

Now consider

[Lz, L²] = [Lz, Lx²] + [Lz, Ly²] + [Lz, Lz²]

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

[Lz, L²] = [Lz, Lx²] + [Lz, Ly²]
and with application of the product property
[Lz, L²] = Lx[Lz, Lz] + [Lz, Lx]Lx + Ly[Lz, Ly] + [Lz, Ly]Ly
then replacing the commutators with
the relations previously developed
[Lz, L²] = ih([LxLy + LyLxLyLxLxLy) = 0

Likewise [Lx, L²]=0 and [Ly, L²]=0.

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

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