Chemistry 460
Spring 2015
Dr. Jean M. Standard
March 25, 2015
Angular Momentum Properties
Classical Definition of Angular Momentum
!
In classical mechanics, the angular momentum vector L is defined as
! ! !
L = r × p ,
(1)
!
!
where r corresponds to the position vector and p is the (linear) momentum vector. The cartesian components of
the angular momentum vector, Lx , Ly , and Lz , can be expressed in terms of the components of position and
momentum using the definition of the cross product,
Lx = ypz − zpy
(2)
Ly = zpx − xpz
(3)
Lz = xpy − ypx .
(4)
The square of the magnitude of the angular moment vector, L2 , is defined as for any vector as the sum of the
squares of the components,
L2 = L2x + L2y + L2z .
(5)
The magnitude L of the angular momentum vector is therefore the square root of the relation in Eq. (5),
1/2
L = !" L2x + L2y + L2z #$ .
(6)
Angular Momentum Operators
To construct quantum mechanical operators for angular momentum, the basic rules for constructing operators are
employed: coordinates are transformed into "multiply by" operators, and momenta are transformed into derivative
operators using the relation
pˆ k = −i!
∂
, k = x, y, or z .
∂k
(7)
Using the rules for operator construction, the angular momentum component operators become
" ∂
∂ %
Lˆ x = yˆpˆ z − zˆpˆ y = −i! $ y − z '
∂y&
# ∂z
(8)
" ∂
∂%
Lˆ y = zˆpˆ x − xˆpˆ z = − i! $ z − x '
# ∂x
∂z &
(9)
" ∂
∂ %
Lˆz = xˆpˆ y − yˆpˆ x = − i! $ x − y '.
∂x &
# ∂y
(10)
2
The operator for the square of the angular momentum also may be constructed,
Lˆ2 = Lˆ2x + Lˆ2y + Lˆ2z .
(11)
Note that there is no Lˆ operator in quantum mechanics. Since quantum mechanical operators must be linear
operators, the square root in the classical definition precludes the use of the magnitude L as an operator.
Spherical Polar Coordinates
The angular momentum is closely related to the angular variables θ and φ of the spherical polar coordinate system.
A point ( r, θ , φ ) in the cartesian axis system is shown in Fig. 1.
€
Figure 1. Diagram illustrating the spherical polar coordinates r, q, and h.
The equations relating cartesian coordinates ( x, y, z ) and spherical polar coordinates
€
x = r sin θ cos φ
y = r sin θ sin φ
z = r cos θ .
( r, θ , φ ) are
(12)
€
The ranges of the coordinates are 0 ≤ r ≤ ∞ , 0 ≤ θ ≤ π , and 0 < φ ≤ 2π . Solving for ( r, θ , φ ) in terms of ( x, y, z )
yields
€
€
€
(
r = € x 2 + y 2 + z 2
"
1/2
)
%
'
θ = cos $
2
2
2 1/2 '
$# x + y + z
'&
z
−1 $
(
" y%
φ = tan −1 $ ' .
#x&
)
€
€
(13)
(14)
(15)
3
The angular momentum component operators also may be expressed in terms of spherical polar coordinates,
"
∂
∂ %
Lˆ x = −i! $ −sin φ
− cot θ cos φ '
∂θ
∂φ &
#
(16)
"
∂
∂ %
Lˆ y = − i! $ cos φ
− cot θ sin φ '
∂θ
∂φ &
#
(17)
∂
Lˆz = − i! .
∂φ
(18)
The square of the angular momentum operator also may be expressed in spherical polar coordinates,
& ∂2
∂
1
∂2
Lˆ2 = Lˆ2x + Lˆ2y + Lˆ2z = − ! 2 (( 2 + cot θ
+
∂θ
sin 2 θ ∂φ 2
' ∂θ
)
++ .
*
(19)
Commutators
€
The components of the angular momentum operators do not commute,
[ Lˆx , Lˆy ]
[ Lˆy , Lˆz ]
[ Lˆz , Lˆx ]
= i!Lˆ z
= i!Lˆ
(20)
x
= i!Lˆ y .
However, each of the components commute with the square of the angular momentum,
€
[ Lˆ2 , Lˆx ] = [ Lˆ2 , Lˆy ] = [ Lˆ2 , Lˆz ]
(21)
= 0.
2
ˆ
Since Lˆ and Lˆ z commute, they can possess a set of simultaneous eigenfunctions. However, since Lˆ x and Ly do
2
not commute with Lˆ z , they€cannot possess the same set of eigenfunctions as Lˆ z and Lˆ .
€
€
Angular Momentum Eigenvalue Equations
€ momentum eigenvalue equations are
The angular
€
€
€
€
Lˆ2 Yℓm (θ , φ ) = ! 2 ℓ ( ℓ +1) Yℓm (θ , φ )
(22)
Lˆz Yℓm (θ , φ ) = m! Yℓm (θ , φ ) .
(23)
Here, ℓ and m are integers, with ℓ = 0, 1, 2, … and m = 0, ± 1, ± 2, …, ± ℓ . The integer ℓ is known as the angular
momentum quantum number and m is known as the magnetic, or azimuthal, quantum number. The functions
Yℓm (θ , φ ) are known as spherical harmonics, and they are discussed in more detail below.
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4
Eigenfunctions of Angular Momentum
The eigenfunctions of the angular momentum operators are the spherical harmonics,
Yℓm (θ ,φ )
m
In Eq. (24), the functions Pℓ
€
& ( 2ℓ + 1) ( ℓ − m ) !)1/ 2
m
+
= (
Pℓ ( cos θ ) e imφ .
(' 4 π ( ℓ + m ) !+*
(24)
( cos θ ) are associated Legendre functions,
m
Pℓ
€
( u) =
1
(
2 ℓ!
ℓ
1− u2
)
m /2
d
ℓ+ m
du
ℓ+ m
(u − 1)
2
ℓ
.
(25)
The associated Legendre functions obey the recursion formula
€
m
( 2ℓ + 1) u Pℓ ( u)
m
m
= (ℓ − m + 1) Pℓ+1( u ) + (ℓ + m ) Pℓ−1( u) .
(26)
The first several associated Legendre polynomials are listed in the table below.
€
P00 ( u) = 1
P10 ( u) = u
(
P11( u) = 1− u 2
€
P20 ( u) =
)
1/ 2
€
P21( u) =
€
P22 ( u) =
1
3u 2 − 1
2
( )
3u (1 − u )
3 (1− u 2 )
2
1/ 2
€
€
As with any quantum mechanical eigenfunctions, the spherical harmonic functions are orthonormal,
2π π
∫∫
€
*
Yℓ'm'
(θ, φ ) Yℓm (θ, φ ) sinθ dθ dφ = δℓ 'ℓ δm'm .
(27)
0 0
Raising and Lowering Operators
The angular momentum raising and lowering operators Lˆ+ and Lˆ− are defined as
Lˆ+ = Lˆ x + i Lˆ y
(28)
€
€
Lˆ− = Lˆ x − i Lˆ y .
(29)
In spherical polar coordinates, the raising and lowering operators become
" ∂
∂ %
Lˆ+ = !eiφ $
− i cot θ '
∂φ &
# ∂θ
(30)
" ∂
∂ %
Lˆ− = !e−iφ $ −
− i cot θ ' .
∂φ &
# ∂θ
(31)
5
Some useful commutators involving the raising and lowering operators are
! Lˆ2 ,
"
! Lˆ ,
" +
! Lˆ ,
" z
! Lˆ ,
" z
Lˆ+ #$ = !" Lˆ2 , Lˆ− #$ = 0
Lˆ− #$ = 2! Lˆz
Lˆ+ #$ = ! Lˆ+
Lˆ− #$ = −! Lˆ− .
(32)
The raising and lowering operators Lˆ+ and Lˆ− have the following effects on the eigenfunctions Yℓm (θ ,φ ) ,
1/2
Lˆ+ Yℓm (θ , φ ) = ! "#ℓ ( ℓ +1) − m ( m +1)$% Yℓ,m+1 (θ , φ )
€
€
1/2
Lˆ− Yℓm (θ , φ ) = ! "#ℓ ( ℓ +1) − m ( m −1)$% Yℓ,m−1 (θ , φ ).
(33)
€
(34)
Note that the raising operator raises the magnetic quantum number m by one and the lowering operator lowers m by
one, but the operators have no effect on the angular momentum quantum number ℓ .