Rotations and Angular momentum

32 Pages·2004·0.28 MB·English

rotations.nb: 11/3/04::13:48:13 1
Rotations and Angular momentum
(cid:1) Intro
The material here may be found in
Sakurai Chap 3: 1-3, (5-6), 7, (9-10)
Merzbacher Chap 11, 17.
Chapter 11 of Merzbacher concentrates on orbital angular momentum. Sakurai, and Ch 17 of Merzbacher focus on
angular momentum in relation to the group of rotations. Just as linear momentum is related to the translation group,
angular momentum operators are generators of rotations. The goal is to present the basics in 5 lectures focusing on
1. J as the generator of rotations.
2. Representations of SO 3
3. Addition of angular mo(cid:1)m(cid:2)entum
4. Orbital angular momentum and Y ' s
lm
5. Tensor operators.
(cid:1) Rotations & SO(3)
ü Rotations of vectors
Begin with a discussion of rotations applied to a 3-dimensional real vectorspace. The vectors are described by three real
v
x
numbers, e.g. v= v . The transpose of a vector is vT = v , v , v . There is an inner product defined between two
(cid:3)(cid:5) y (cid:6)(cid:8) x y z
(cid:5) (cid:8)
(cid:5)(cid:5)(cid:5)vz (cid:8)(cid:8)(cid:8) (cid:1) (cid:2)
(cid:5) (cid:8)
vectors by uTÿv=(cid:5)(cid:5)vTÿ(cid:8)(cid:8)u=uv cosf, where f is the angle between the two vectors. Under a rotation the inner product
(cid:4) (cid:7)
between any two vectors is preserved, i.e. the length of any vector and the angle between any two vectors doesn't
change. A rotation can be described by a 3ä3real orthogonal matrix R which operates on a vector by the usual rules of
matrix multiplication
v' v
x x
v' =R v and v' , v' , v' = v , v , v RT
(cid:3)(cid:5) y (cid:6)(cid:8) (cid:3)(cid:5) y(cid:6)(cid:8) x y z x y z
(cid:5) (cid:8) (cid:5) (cid:8)
(cid:5)(cid:5)(cid:5) v'z (cid:8)(cid:8)(cid:8) (cid:5)(cid:5)(cid:5)vz (cid:8)(cid:8)(cid:8) (cid:1) (cid:2) (cid:1) (cid:2)
(cid:5) (cid:8) (cid:5) (cid:8)
(cid:5)(cid:5) (cid:8)(cid:8) (cid:5)(cid:5) (cid:8)(cid:8)
(cid:4) (cid:7) (cid:4) (cid:7)
To preserve the inner product, it is requird that RTÿR =1
u'ÿv'=uRTÿRv
=u1v
=uÿv
As an example, a rotation by f around the z-axis (or in the xy-plane) is given by
rotations.nb: 11/3/04::13:48:13 2
cosf -sinf 0
R f = sinf cosf 0
z (cid:3)(cid:5) (cid:6)(cid:8)
(cid:5) (cid:8)
(cid:1) (cid:2) (cid:5)(cid:5)(cid:5) 0 0 1(cid:8)(cid:8)(cid:8)
(cid:5) (cid:8)
(cid:5)(cid:5) (cid:8)(cid:8)
(cid:4) (cid:7)
The sign conventions are appropriate for a right handed coordinate system: put the thumb of right hand along z-axis,
extend fingers along x-axis, and curl fingers in direction of y-axis.
z
y
x
The direction of rotation for f is counter-clockwise when looking down from the +zdirection, i.e. rotate the x-axis into
the y-axis. Similarly the rotations around the x and y axes are
cosf 0 sinf 1 0 0
R f = 0 1 0 and R f = 0 cosf -sinf
y (cid:3)(cid:5) (cid:6)(cid:8) x (cid:3)(cid:5) (cid:6)(cid:8)
(cid:5) (cid:8) (cid:5) (cid:8)
(cid:1) (cid:2) (cid:5)(cid:5)(cid:5)-sinf 0 cosf(cid:8)(cid:8)(cid:8) (cid:1) (cid:2) (cid:5)(cid:5)(cid:5)0 sinf cosf (cid:8)(cid:8)(cid:8)
(cid:5) (cid:8) (cid:5) (cid:8)
(cid:5)(cid:5) (cid:8)(cid:8) (cid:5)(cid:5) (cid:8)(cid:8)
(cid:4) (cid:7) (cid:4) (cid:7)
The sign of sinf in R is related to the handed-ness of the coordinate system and the sense of rotation. For
y
3-dimensions, it is equivalent to talk about rotations around the z-axis, or rotations in the xy-plane. In any other number
of dimensions, the correct language is to talk about rotations in the x x -plane, where x defines one of the coordinate
i j i
directions of the vector space. Thus, while for 3-dimensions there are 3 independent rotations, in N-dimensions, there
will be ÅNÅÅÅÅÅNÅÅÅÅÅ-ÅÅ1ÅÅÅÅÅÅ independent rotations.
(cid:1)2 (cid:2)
ü Direction kets
To make the correspondence to quantum states, just as a translation was defined by its action on position eigenkets, a
rotation around the origin also can act on position eigenkets by
è
x '=R x = Rx
(cid:9) (cid:10) (cid:9) (cid:10) (cid:9) (cid:10) è
where a distinction has been made between the operator R, which acts on the state, and the rotation matrix R which acts
on the coordinates. Since the rotations don't change the length of the vector, it is possible to define spherical
` `
coordinates, r, q, f, and spherical position kets, x Ø r ≈ n , where r determines the radial position, and n indicates
`
the direction from the origin. The rotations act on(cid:9)ly(cid:10) on(cid:1) th(cid:2)e (cid:1)n(cid:2) degrees of freedom.
è ` ` (cid:9) (cid:10)
R r ≈ n = r ≈ Rÿn
(cid:1)(cid:9) (cid:10) (cid:9) (cid:10)(cid:2) (cid:9) (cid:10) (cid:9) (cid:10)
Direction kets will be used more extensively in the discussion of orbital angular momentum and spherical harmonics,
`
but for now they are useful for illustrating the set of rotations. The set of all direction kets n can be visualized by the
surface of a sphere, and the rotations are the set of all possible ways to reorient that sphere.(cid:11) (cid:9) (cid:10)(cid:12)
rotations.nb: 11/3/04::13:48:13 3
ü Orthognal group SO(3)
The set of all possible rotations form a group. Consider the four properties: closure, identity, inverse and associativity.
Using the picture of rotations as reorientations of a sphere, one can construct visualizations to illustrate each property.
With greater mathematical rigor, the set of all possible rotations form the group SO(3), where O Ø orthogonal, 3 Ø 3
dimensions, and S Ø special, which in this case means the matrix has a determinant of 1. The rotations are described by
three continuous, but bounded, parameters. From the matrix point of view, a 3ä3matrix has nine degrees of freedom.
The constraint that the matrix is orthogonal, RT R =d yields 6 conditions, i.e. three for i=k and three for i∫k.
ij jk ik
The properties of a group are obeyed:
closure: For any two orthogonal matrices R and R , the product R =R R , is also orthogonal. The combination
1 2 3 1 2
of two rotations is also a rotation.
identity: The 3ä3 unit matrix acts as an identity element for the group. 1 R =R1=R
inverse: Each element has an inverse R-1 =RT, RT R=RRT =1
associativity: R R R = R R R
1 2 3 1 2 3
(cid:1) (cid:2) (cid:1) (cid:2)
Unlike the Translation group, SO(3) is not abelian, i.e. in general R R ∫R R .
1 2 2 1
The significance of the S-condition, DetR=1, is that reflections are not included in the group, i.e. for three dimensions
one cannot turn a right-handed object into a left-handed object by doing a rotation. If we allowed reflections, e.g.
-1 0 0
0 -1 0
(cid:3) (cid:6)
(cid:5) (cid:8)
(cid:5) (cid:8)
(cid:5)(cid:5) 0 0 -1 (cid:8)(cid:8)
(cid:5) (cid:8)
(cid:5) (cid:8)
(cid:5)(cid:5) (cid:8)(cid:8)
(cid:4) (cid:7)
Then, the group would be O(3) instead of SO(3). O(3) is called "disconnected" since not all elements of the group can
be reached by a succession of infinitesimal transformations. SO(3) is connected.
The rotation matrices R are just one "representation" of the group SO(3). For two different representations, there has to
be a 1¨1 mapping of the elements of one representation to the other. The mapping has to preserve the combination law.
Consider two representations R and S. Label a rotation by a subscript which represents the three parameters to define a
rotation, and identify R ¨S , If R =R R , then we must have S =S S to preserve the combinatin law.
a a 3 1 2 3 1 2
`
ü Full set of rotations: (cid:1)n, f(cid:2)
There are two common methods for parameterizing rotations. The first is to choose an axis for rotation and then perform
a rotation by an angle between 0 and p. The axis of rotation can be chosen anywhere on the sphere. Why not 0¨2 p ?
`
Then rotations with poles on opposite sides of the sphere would be redundant. An explicit form for R n, f will be given
after developing the language of infinitesimal rotations. (cid:1) (cid:2)
Draw yourownpictureshowingtherotation ofaspherearoundanoff(cid:1)
(cid:2)
axispole.Thesphererepresentsthesetofstates, n,
i.e.thesetofdirectionkets.Therotationreorientsthesphere.
rotations.nb: 11/3/04::13:48:13 4
ü Euler angles
The second parameterization is to give Euler angles. In this method one describes where the "north pole" moves to
under a rotation, and the orientation of the sphere after the pole has been moved. The location of the pole is determined
by first choosing a longitude by rotating around the z-axis, then a latitude by rotating around the new y-axis. Finally,
the orientation of the sphere is given by a final rotation around the new z-axis. Pictorially,
Drawmorepictures, showingthesequenceofrotationstomovethepole,
andthenreorientthespherearoundthe newpole.
In the Euler parameterization the range of angles is
a= 0, 2 p , b= 0, p , g= 0, 2 p
(cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2)
and an arbitrary rotation is given by
R a, b, g =R g R b R a
z' y' z
(cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2)
Note that the z' and y' rotations are not defined with respect to the original coordinate axes, but rather with respect to
where those axes have moved with the reorientation of the sphere. Later it will be shown that
R g R b R a =R a R b R g
z' y' z z y z
(cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2)
where the order has been reversed, but now all rotations are conveniently defined around the axes of the original
coordinate system.
ü Equivalency of the two parameterizations
`
The two parameterizations may not seem equivalent, but they are, as can be seen by a pictorial mapping of n, f to
a, b, g . Observe that there are two ways to produce the same set of Euler angles consistent with the restri(cid:1)ction(cid:2) of f to
(cid:1)0, p . (cid:2)
(cid:1) (cid:2)
Thispicturedidn'(cid:3)tmakeitintotheclassroompresentation.it'(cid:3)sabitofwork
The two techniques have different uses. Euler angles tend to be more useful for building up actual rotation matrices in a
calculation. This is because R and R are generally fairly easy to construct for a representation, and the matrix
z y
`
multiplication is straightforward. The n, f notation has advantages in some analytic manipulations, as we will see
below. (cid:1) (cid:2)
rotations.nb: 11/3/04::13:48:13 5
ü J as the generator of infinitesimal rotations.
In analogy to the discussion of translations and time evolution, it is useful to build up the finite rotations from
generators of infinitesimal rotations. Recognizing that we are eventually interested in a quantum mechanical
formulation, it is useful to develop this formalism in a way that realizes the rotations as unitary operations. For example,
an infinitesimal rotation around the z-axis is given by
R d =1-idJ
z z
(cid:1) (cid:2)
where J is the generator of infinitesimal rotations around the z-axis. Since R is unitary (note: orthogonal matrices are
z
unitary), J must be Hermitian. In the present case, to leading order in d
1 -d 0 0 -i 0
R d = d 1 0 or J = i 0 0
z (cid:3)(cid:5) (cid:6)(cid:8) z (cid:3)(cid:5) (cid:6)(cid:8)
(cid:5) (cid:8) (cid:5) (cid:8)
(cid:1) (cid:2) (cid:5)(cid:5)(cid:5)0 0 1(cid:8)(cid:8)(cid:8) (cid:5)(cid:5)(cid:5)0 0 0(cid:8)(cid:8)(cid:8)
(cid:5) (cid:8) (cid:5) (cid:8)
(cid:5)(cid:5) (cid:8)(cid:8) (cid:5)(cid:5) (cid:8)(cid:8)
(cid:4) (cid:7) (cid:4) (cid:7)
Similarly, for this representation
0 0 0 0 0 i
J = 0 0 -i , J = 0 0 0
x (cid:3)(cid:5) (cid:6)(cid:8) y (cid:3)(cid:5) (cid:6)(cid:8)
(cid:5) (cid:8) (cid:5) (cid:8)
(cid:5)(cid:5)0 i 0 (cid:8)(cid:8) (cid:5)(cid:5)-i 0 0(cid:8)(cid:8)
(cid:5) (cid:8) (cid:5) (cid:8)
(cid:5) (cid:8) (cid:5) (cid:8)
(cid:5)(cid:5) (cid:8)(cid:8) (cid:5)(cid:5) (cid:8)(cid:8)
(cid:4) (cid:7) (cid:4) (cid:7)
Note that since this is still a classical discussion I haven't put in any factors of (cid:1).
ü Commutation relations
The generators obey the commutation relations
J, J =ie J
i j ijk k
(cid:13) (cid:14)
where e =1, if i jk isanevenpermutationof xyz
ijk
=-1, if (cid:1)i jk(cid:2)isanoddpermutationof x(cid:1) yz (cid:2)
=0, if a(cid:1)nytw(cid:2)oof i jk areequal (cid:1) (cid:2)
(cid:1) (cid:2)
as a useful aside e e =d d -d d .
ijk lmk il jm im jl
The commutation relations are a property of the group, not just a particular representation. The collection of all the
commutator relations for the generators is sometimes called the algebra of the generators of the group, or just the
algebra of the group.
ü Finite rotations
For rotations around a particular axis, it should be clear that we can build up an arbitrary rotation by a sequence of
infinitesimal rotations, similar to the procedure for building up a finte translation as the limiting product of a large
number of infinitesimals.
rotations.nb: 11/3/04::13:48:13 6
R f =Lim PR f n
z
nض i
(cid:1) (cid:2) (cid:15) (cid:1) (cid:16) (cid:2)(cid:17)
= Lim 1-iJ f n n
z
nض
=e-iJzf(cid:1) (cid:1) (cid:16) (cid:2)(cid:2)
`
It is now possible to give a form for an arbitrary rotation in the n, f parameterization. The infinitesimal rotation
`
around the n-axis is given by (cid:1) (cid:2)
` `
R n, d =1-idnÿJ
(cid:1) (cid:2)
There is no concern about which component of J is operated on first, since the effects of commutation amongst the
different genreators shows up at second order in the infinitesimal d. The finite rotation is then given by
` `
R n, f =e-ifnÿJ
(cid:1) (cid:2)
(cid:1) Representations of SO(3)
ü relation between D and R - notational
Rotations can act to change a wide variety of objects, e.g. classical position vectors, position eigenstates x , operators
such as X , P or L, angular momentum states lm etc. In principle, the notation R to denote a rotation op(cid:9)e(cid:18)r(cid:10)ator can be
(cid:19)(cid:19)(cid:19)(cid:18) (cid:19)(cid:18) (cid:19)(cid:18)
used for all of these applications, if a sufficien(cid:9)tly(cid:10) detailed definition is supplied for the case at hand. In practice, a
common convention is to use R when the object in question has the properties of a position vector, but to use a notation
D when operating on angular momentum states or objects with similar characteristics. For example, to operate on a
classical vector use
x' =R x
i ij j
or to operate on a position eigenstate
x' =R x = R x
ij j
(cid:9) (cid:10) (cid:9) (cid:10) (cid:9) (cid:10)
In contrast, to perform a rotation on a state a which is an angular momentum state a = lm one would write
(cid:9) (cid:10) (cid:9) (cid:10) (cid:9) (cid:10)
a' =D R lm
(cid:20) (cid:2) (cid:1) (cid:2)(cid:20) (cid:2)
where D R is an operator that depends on orbital angular momentum l. One would still write D in the exponential form
(cid:1) (cid:2)
D n`, f =e-Å(cid:1)ÅiÅÅ Jÿn`f
(cid:1) (cid:2)
but the form of the generators would be specific to the set of states, or representation, which is the object of the rotation.
The set of possible representations is quite large. To simplify the discussion as much as possible, one defines the
irreducible representations of SO 3 which is the subject of the next section.
(cid:1) (cid:2)
rotations.nb: 11/3/04::13:48:13 7
ü Irreducible representations
ü Casimir operators and maximal set of commuting operators.
The first item of business is to determine a maximal set of commuting operators that can be simultaneaously
diagonalized. For SO(3) or SU(2) this would be J2 and one other component of J, typically taken to be J . J2 is an
z
example of what is known as a Casimir operator. Casimir operators commute with all operators within the algebra of the
group. Other groups may have more than one Casimir. The number of Casimir operators is equal to the "rank" of the
group. SO(4) and SU(3) for example have two Casimir operators and are rank 2. Generally, the most useful Casimir is
the quadratic operator C2 = O2 where the O are the infinitesimal generators of the group. J2 is such a quadratic
i i i
casimir operator. In groups w(cid:21)ith more than one casimir operator, they all commute with each other.
In addition to the casimir operators, one can choose a set of operators equal in number to the rank from the group
algebra that also commute with each other. For example in SU(3) one can find two generators that are diagonal. So in
general, the dimension of a maximal set of commuting operators with which to define the representations of a group is
twice the rank.
For completeness,
J2, J = J2+J2+J2, J
z x y z z
(cid:13) (cid:14) =(cid:13)J2+J2, J (cid:14)
x y z
=(cid:13)J J , J +(cid:14)J , J J +J J , J + J , J J
x x z x z x y y z y z y
=-i(cid:13)J J (cid:14)+J(cid:13) J +(cid:14)i J J +(cid:13)J J (cid:14) (cid:13) (cid:14)
x y y x y x x y
=0 (cid:1) (cid:2) (cid:1) (cid:2)
ü Labeling of states
Since J2, J can be diagonalized simultaneously, we can specifiy the states by j, m , where m is the eignevalue when
z
Jz operates on the state and the operation of J2 yields aj (cid:9) (cid:10)
J j, m =m j, m
z
J2(cid:20) j, m(cid:2) =aj(cid:20) j, m(cid:2)
(cid:20) (cid:2) (cid:20) (cid:2)
where the eigenvalue of J2 is not yet determined. Note that the labeling of the states is rather arbitrary. In the case of J ,
z
it is convenient to use the eigenvalue directly. We will also take the states j, m to be normalized.
(cid:9) (cid:10)
ü j-representations as irreducible representations
As J2 commutes with all the generators of the group, it also commutes with any function of those generators, and in
particular J2, D =0. As with other commutation relations, this implies that the rotation operators don't change the
eigenvalue(cid:13) of J2(cid:14),
J2 D jm =DJ2 jm =Da jm =a D jm
j j
(cid:1) (cid:9) (cid:10)(cid:2) (cid:9) (cid:10) (cid:9) (cid:10) (cid:1) (cid:9) (cid:10)(cid:2)
rotations.nb: 11/3/04::13:48:13 8
On the other hand J , D ∫0, and so the rotations mix states of different m-value but not of different j-value. In this
z
case we say that al(cid:13)l the s(cid:14)tates of a given j-value, taken together, form a representation of the group. The dimension of
the representation is equal to the number of distinct basis states which may be chosen for the same j-value. In the case
of j-representations we say that the representation is irreducible, i.e. it is not possible to break the representation down
into two subspaces that don't mix under the action of rotations. Thus, the result of performing a rotation on a state jm
is given by a linear combination of all states jm' (cid:9) (cid:10)
(cid:9) (cid:10)
D jm =S Dj jm'
mm'
m'
(cid:9) (cid:10) (cid:9) (cid:10)
where the exact value of the coefficients Dj depends on the parameters describing the rotation.
mm'
(cid:1) Orbital angular momentum
This section develops orbital angular momentum operators in a manner analagous to the development of momentum as
the generator of translations.
ü Direction kets
`
Begin with the direction kets n , or q, f . The angular behavior of a particle in a given state, say b , is given by
`
yb q, f = n b . This is in d(cid:9)ire(cid:10)ct an(cid:9)alog(cid:10)y to the spatial wave function yb x = x b . Instead of (cid:9)q,(cid:10)f one can use
`
z=(cid:1)cos(cid:2)q as(cid:22) th(cid:20)e p(cid:10)olar coordinate, and states z, f , so that yb z, f = n b(cid:1).(cid:2) (cid:22) (cid:20) (cid:10) (cid:9) (cid:10)
(cid:9) (cid:10) (cid:1) (cid:2) (cid:22) (cid:20) (cid:10)
ü Rotations
` ` `
The rotation acting on the direction ket gives n' =D a, b, g n = R a, b, g ÿn , where here R is reserved to denote
the rotation of vectors, D to denote the transfo(cid:9)rm(cid:10)ation(cid:1) of state(cid:2)s(cid:9), (cid:10)and (cid:9)th(cid:1)e rotatio(cid:2)n i(cid:10)s labeled by the Euler angles. This is
analagous to x' =T D x = x+D .
(cid:9) (cid:10) (cid:1) (cid:2)(cid:9) (cid:10) (cid:9) (cid:10)
ü Infinitesimal Rotations
The generators of R were defined above as 3ä3 matrices J . Let the corresponding genertors to operate on states be
x,y,z
` ` `
labeled L . For example, they act on direction kets by n Ø n' (cid:1) 1-idnÿL n .
x,y,z
(cid:9) (cid:10) (cid:9) (cid:10) (cid:1) (cid:2)(cid:9) (cid:10)
ü L as a differential operator
For spatial translations, the differential form of the momentum operator was uncovered by showing that
x p b =-i(cid:1) ÅÅ∑ÅÅÅÅÅ x b . For L the analagous relation is
∑x z
(cid:22) (cid:23) (cid:9) (cid:10) (cid:22) (cid:20) (cid:10)
` `
n L b =-i(cid:1) ÅÅ∑ÅÅÅÅÅ n b
z ∑f
`
(cid:22) (cid:23) (cid:9) (cid:10)=-i(cid:1) x ÅÅ∑ÅÅÅÅÅ(cid:22)-(cid:20)y Å(cid:10)Å∑ÅÅÅÅÅ n b
∑y ∑x
`
= x p(cid:24) -x p n(cid:25)(cid:22)b(cid:20) (cid:10)
x y y x
(cid:1) (cid:2)(cid:22) (cid:20) (cid:10)
More generally, a rotation around the k-axis (or in the ij-plane) is given by
` `
n L b =e x p -x p n b
k ijk i j j i
(cid:22) (cid:23) (cid:9) (cid:10) (cid:1) (cid:2)(cid:22) (cid:20) (cid:10)
rotations.nb: 11/3/04::13:48:13 9
A little effort shows that the algebra of the generators L is
L, L =ie L
i j ijk k
(cid:13) (cid:14)
The generators can be written explicitly in terms of angular variables in the usual coordinates
L Ø-i(cid:1) -sinf ÅÅ∑ÅÅÅÅÅ -cotqcosf ÅÅ∑ÅÅÅÅÅ
x ∑q ∑f
L Ø-i(cid:1)(cid:24)cosf ÅÅ∑ÅÅÅÅÅ -cotqsinf ÅÅ∑ÅÅÅÅÅ (cid:25)
y ∑q ∑f
L Ø-i(cid:1) (cid:24)ÅÅ∑ÅÅÅÅÅ (cid:25)
z ∑f
These can be combined to yield the Casimir operator L2
L2 Ø-(cid:1)2 ÅÅÅÅÅ1ÅÅÅÅÅÅÅÅ ÅÅ∑ÅÅÅ2ÅÅÅÅ + ÅÅÅÅ1ÅÅÅÅÅÅ ÅÅ∑ÅÅÅÅÅ sinq ÅÅ∑ÅÅÅÅÅ
sin2 q ∑2f sinq ∑q ∑q
(cid:24) (cid:25)
It is not a coincidence that L2 and L are the differential operators that arise from performing the separation of
z
variables. Although I haven't proven it myself, I suspect strongly that there is an equivalence between the ability to
separate variables and the ability to find a maximal set of commuting variables.
ü Eigenstates of L2, L
z
Eigenstates of angular momentum can be chosen to be simultaneous eigenvalues of J2 and J . For orbital angular
z
momentum we expect eigenstates of L2 and L . Suppose, therefore, that b is an eigenstate of L and L2 with
z z
eigenvalues m and l l+1 respectively. Then labeling the state by lm gi(cid:9)ve(cid:10)s
(cid:1) (cid:2) (cid:9) (cid:10)
` ` `
n L lm =-i(cid:1) ÅÅ∑ÅÅÅÅÅ n lm =m n lm
z ∑f
(cid:22) (cid:23) (cid:9) (cid:10) (cid:22) (cid:20) (cid:10) (cid:22) (cid:20) (cid:10)
This equation is independent of q, so it is reasonable to separate variables in both the coordinates and the states
`
n = z, f = z ä f
(cid:9)lm(cid:10) =(cid:9) l m(cid:10)ä m(cid:9) (cid:10) (cid:9) (cid:10)
(cid:9) (cid:10) (cid:9) (cid:10) (cid:9) (cid:10)
where m -kets exist in the f -space, and l -kets exist in the z or q -space. The notation l suggests that for each
m m
`
value o(cid:9)f m(cid:10) , there are a set o(cid:9)f s(cid:10)tates labeled(cid:9) (cid:10)by l. The allowed ra(cid:1)n(cid:2)ge o(cid:9)f (cid:10)l is l ¥ m . The ampli(cid:9)tu(cid:10)de n lm separates as
well (cid:9) (cid:23) (cid:22) (cid:20) (cid:10)
`
n lm = z l f m
m
(cid:22) (cid:20) (cid:10) (cid:22) (cid:20) (cid:10) (cid:22) (cid:20) (cid:10)
The differential equation can then be rewritten by dividing through by z l ,
m
(cid:22) (cid:20) (cid:10)
-i(cid:1) ÅÅ∑ÅÅÅÅÅ f m =m f m
∑f
(cid:22) (cid:20) (cid:10) (cid:22) (cid:20) (cid:10)
and solved
f m = ÅÅÅÅÅ1ÅÅÅÅÅÅÅÅÅ eimf
2 p
(cid:3)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)
(cid:22) (cid:20) (cid:10)
where the normalization is chosen so that
rotations.nb: 11/3/04::13:48:13 10
m m' = m 1 m'
(cid:22) (cid:20) (cid:10)=(cid:22)m(cid:23) (cid:9)„f(cid:10) f f m'
=(cid:22) „(cid:23)f(cid:1)(cid:26) m f(cid:9) (cid:10) (cid:22)f (cid:23)(cid:2)m(cid:9)' (cid:10)
=(cid:26)2 p„(cid:22)f ÅÅ(cid:20)Å1ÅÅÅÅÅ(cid:10) e(cid:22)im'(cid:20)-m f(cid:10)
0 2 p (cid:1) (cid:2)
=d(cid:26)
mm'
Similarly, in the z-space, z l is a function only of q. In the differential equation for L2 one can substitute the
m
solution for Lz (cid:22) (cid:20) (cid:10)
n` L2 lm =-(cid:1)2 ÅÅÅÅÅ1ÅÅÅÅÅÅÅÅ ÅÅ∑ÅÅÅ2ÅÅÅÅ + ÅÅÅÅ1ÅÅÅÅÅÅ ÅÅ∑ÅÅÅÅÅ sinq ÅÅ∑ÅÅÅÅÅ n` lm
sin2 q ∑2f sinq ∑q ∑q
(cid:22) (cid:23) (cid:9) (cid:10) (cid:24) (cid:25)(cid:22) (cid:20) (cid:10)
or
z L2 l =-(cid:1)2 ÅÅ-ÅÅÅmÅÅÅÅÅ2ÅÅÅÅ + ÅÅÅÅ1ÅÅÅÅÅÅÅ ÅÅ∑ÅÅÅÅÅ sinf ÅÅ∑ÅÅÅÅÅ z l
m sin2 f sinf ∑q ∑q m
(cid:22) (cid:23) (cid:9) (cid:10) (cid:24) (cid:25)(cid:22) (cid:20) (cid:10)
which has for solutions the associated Legendre polynomials.
z l =c Pm z
m lm l
(cid:22) (cid:20) (cid:10) (cid:1) (cid:2)
The normalization is given by the requirement
l l =1
m m
(cid:22) (cid:20) (cid:10) = „z ml z z l m
=(cid:26)„zc(cid:22)lm (cid:20)cl*m(cid:10)(cid:22)Plm(cid:20) (cid:10)z 2
=(cid:26)c2 ÅÅÅÅÅ2ÅÅÅÅÅÅÅÅ ÅÅlÅÅ+ÅÅ(cid:1)ÅÅmÅÅÅÅÅÅÅ!ÅÅÅ(cid:1) (cid:2)(cid:2)
lm 2 l+1 (cid:1)l-(cid:5)m(cid:6)(cid:2)!
(cid:1) (cid:5) (cid:6)(cid:2)
or, with a conventional choice of phase, c = - m Å2ÅÅ ÅlÅ+ÅÅÅÅÅ1ÅÅÅ ÅÅlÅÅ-ÅÅÅÅmÅÅÅÅÅÅÅ!ÅÅÅ
lm 2 (cid:1)l+(cid:5)m(cid:6)(cid:2)!
(cid:27)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)
(cid:1) (cid:2) (cid:1) (cid:5) (cid:6)(cid:2)
Combining the two forms, gives the spherical harmonics
n` lm =Y q, f = - m Å2ÅÅ ÅlÅ+ÅÅÅÅÅ1ÅÅÅ ÅÅÅlÅ-ÅÅÅÅÅmÅÅÅÅÅÅ!ÅÅÅ eimf Pm cosq
lm 4 p (cid:1)l+(cid:5)m(cid:6)(cid:2)! l
(cid:27)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)(cid:28)
(cid:22) (cid:20) (cid:10) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:5) (cid:6)(cid:2) (cid:1) (cid:2)
ü Orthogonality and completeness
`
There are a number of orthogonality and completeness relations regarding the states lm and the direction-kets n .
Start with (cid:9) (cid:10) (cid:9) (cid:10)
` ` ` `
n n' =d n, n'
` `
(cid:22)1=(cid:20) (cid:10)„W' (cid:1)n' n(cid:2)'
(cid:26) (cid:9) (cid:10)(cid:22) (cid:23)
so that
` ` `
y q, f = n „W' n' n' b
b
` ` `
(cid:1) (cid:2)= (cid:22)„(cid:23)W(cid:1)(cid:26)' n n(cid:9)' (cid:10)n(cid:22)' (cid:23)b(cid:2)(cid:9) (cid:10)
` ` `
=(cid:26)sinq(cid:22)' „(cid:20)q' „(cid:10)f(cid:22)' d(cid:20)n,(cid:10)n' n' b
=y(cid:26) b q, f (cid:1) (cid:2)(cid:22) (cid:20) (cid:10)
(cid:1) (cid:2)
To make the last step valid, the d-function needs to be defined as