﻿ Angular Momentum in Quantum Mechanics
San José State University

applet-magic.com
Thayer Watkins
Silicon Valley
U.S.A.

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

• 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

## 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]pxand 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

## The Commutivity of Angular Momentum Operators

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

Let

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]Lythen replacing the commutators withthe relations previously developed [Lz, L²] = ih([LxLy + LyLx − LyLx − LxLy) = 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.