Quantum Mechanical Addition of Angular Momenta and Spin

42 Pages·1999·0.34 MB·English

Chapter 6
Quantum Mechanical Addition of
Angular Momenta and Spin
In this section we consider composite systems made up of several particles, each carrying orbital
angularmomentumdecribedbysphericalharmonicsY (θ,φ)aseigenfunctionsand/orspin. Often
(cid:96)m
thesocalledtotal angular momentum,classicallyspeakingthesumofallangularmomentaandspins
of the composite system, is the quantity of interest, since related operators, sums of orbital angular
momentum and of spin operators of the particles, commute with the Hamiltonian of the composite
system and, hence, give rise to good quantum numbers. We like to illustrate this for an example
involving particle motion. Further below we will consider composite systems involving spin states.
Example: Three Particle Scattering
Consider the scattering of three particles A, B, C governed by a Hamiltonian H which depends
only on the internal coordinates of the system, e.g., on the distances between the three particles,
but neither on the position of the center of mass of the particles nor on the overall orientation of
the three particle system with respect to a laboratory–fixed coordinate frame.
To specify the dependency of the Hamiltonian on the particle coordinates we start from the nine
numbers which specify the Cartesian components of the three position vectors (cid:126)r , (cid:126)r , (cid:126)r of the
A B C
particles. Since the Hamiltonian does not depend on the position of the center of mass R(cid:126) =
(m (cid:126)r +m (cid:126)r +m (cid:126)r )/(m +m +m ), sixparametersmustsufficetodescribetheinteraction
A A B B C C A B C
of the system. The overall orientation of any three particle configuration can be specified by
three parameters1, e.g., by a rotational vector ϑ(cid:126). This eleminates three further parameters from
the dependency of the Hamiltonian on the three particle configuration and one is left with three
parameters. How should they be chosen?
Actually there is no unique choice. We like to consider a choice which is physically most reasonable
in a situation that the scattering proceeds such that particles A and B are bound, and particle C
impinges on the compound AB coming from a large distance. In this case a proper choice for a
description of interactions would be to consider the vectors (cid:126)r = (cid:126)r −(cid:126)r and ρ(cid:126) = (m (cid:126)r +
AB A B C A A
m (cid:126)r )/(m +m )−(cid:126)r , and to express the Hamiltonian in terms of |(cid:126)r |, |ρ(cid:126) |, and (cid:94)((cid:126)r ,ρ(cid:126) ).
B B A B C AB C AB C
The rotational part of the scattering motion is described then in terms of the unit vectors rˆ and
AB
1We remind the reader that, for example, three Eulerian angles α,β,γ are needed to specify a general rotational
transformation
141
142 Addition of Angular Momenta and Spin
ρˆ , each of which stands for two angles. One may consider then to describe the motion in terms
C
of products of spherical harmonics Y (rˆ )Y (ρˆ ) describing rotation of the compound AB
(cid:96)1m1 AB (cid:96)2m2 C
and the orbital angular momentum of C around AB.
One can describe the rotational degrees of freedom of the three-particle scattering process through
the basis
B = {Y (rˆ )Y (ρˆ ),(cid:96) = 0,1,...,(cid:96) ,−(cid:96) ≤ m ≤ (cid:96) ;
(cid:96)1m1 AB (cid:96)2m2 C 1 1,max 1 1 1
(cid:96) = 0,1,...,(cid:96) ,−(cid:96) ≤ m ≤ (cid:96) } (6.1)
2 2,max 2 1 2
where (cid:96) and (cid:96) denote the largest orbital and rotational angular momentum values, the
1,max 2,max
values of which are determined by the size of the interaction domain ∆ , by the total energy E,
V
by the masses mA, mB, mC, and by the moment of inertia IA−B of the diatomic molecule A–B
approximately as follows
∆ 2m m m E 1
A B C
(cid:96)1,max = (cid:126) (cid:114)m m + m m + m m , (cid:96)2,max = (cid:126) (cid:112)2IA−BE . (6.2)
A B B C A C
The dimension d(B) of B is
(cid:96)1,max (cid:96)2,max
d(B) = (2(cid:96) + 1) (2(cid:96) + 1) = ((cid:96) + 1)2((cid:96) + 1)2 (6.3)
(cid:88) 1 (cid:88) 2 1,max 2,max
(cid:96)1=0 (cid:96)2=0
For rather moderate values (cid:96) = (cid:96) = 10 one obtains d(B) = 14 641, a very large number.
1,max 2,max
Suchlargenumberofdynamicallycoupledstateswouldconstituteaseriousprobleminanydetailed
description of the scattering process, in particular, since further important degrees of freedom, i.e.,
vibrations and rearrangement of the particles in reactions like AB + C → A + BC, have not
even be considered. The rotational symmetry of the interaction between the particles allows one,
however, to separate the 14 641 dimensional space of rotational states Y (rˆ )Y (ρˆ ) into
(cid:96)1m1 AB (cid:96)2m2 C
subspaces B , B ⊕B ⊕... = B such, that only states within the subspaces B are coupled in the
k 1 2 k
scattering process. In fact, as we will demonstrate below, the dimensions d(B ) of these subspaces
k
does not exceed 100. Such extremely useful transformation of the problem can be achieved through
the choice of a new basis set
B(cid:48) = { c(n) Y (rˆ )Y (ρˆ ), n = 1,2,...14 641} . (6.4)
(cid:88) (cid:96)1,m1;(cid:96)2,m2 (cid:96)1m1 AB (cid:96)2m2 C
(cid:96)1,m1
(cid:96)2,m2
The basis set which provides a maximum degree of decoupling between rotational states is of great
principle interest since the new states behave in many respects like states with the attributes
of a single angular momentum state: to an observer the three particle system prepared in such
states my look like a two particle system governed by a single angular momentum state. Obviously,
composite systems behaving like elementary objects are common, albeit puzzling, and the following
mathematicaldescriptionwillshedlightontheirubiquitousappearenceinphysics,infact,willmake
their appearence a natural consequence of the symmetry of the building blocks of matter.
There is yet another important reason why the following section is of fundamental importance for
the theory of the microscopic world governed by Quantum Mechanics, rather than by Classical
Mechanics. The latter often arrives at the physical properties of composite systems by adding the
6.0: Addition of Angular Momenta and Spin 143
correspondingphysicalpropertiesoftheelementarycomponents; examplesarethetotalmomentum
or the total angular momentum of a composite object which are the sum of the (angular) momenta
of the elementary components. Describing quantum mechanically a property of a composite object
as a whole and relating this property to the properties of the elementary building blocks is then the
quantum mechanical equivalent of the important operation of addittion. In this sense, the reader
will learn in the following section how to add and subtract in the microscopic world of Quantum
Physics, presumably a facility the reader would like to acquire with great eagerness.
Rotational Symmetry of the Hamiltonian
As pointed out already, the existence of a basis (6.4) which decouples rotational states is connected
with the rotational symmetry of the Hamiltonian of the three particle system considered, i.e.,
connected with the fact that the Hamiltonian H does not depend on the overall orientation of the
threeinteractingparticles. Hence, rotationsR(ϑ(cid:126))ofthewavefunctionsψ((cid:126)r ,ρ(cid:126) )definedthrough
AB C
R(ϑ(cid:126))ψ((cid:126)r ,ρ(cid:126) ) = ψ(R−1(ϑ(cid:126))(cid:126)r , R−1(ϑ(cid:126))ρ(cid:126) ) (6.5)
AB C AB C
(cid:48)
do not affect the Hamiltonian. To specify this property mathematically let us denote by H the
(cid:48)
Hamiltonian in the rotated frame, assuming presently that H might, in fact, be different from H.
It holds then H(cid:48)R(ϑ(cid:126))ψ = R(ϑ(cid:126))Hψ. Since this is true for any ψ((cid:126)r ,ρ(cid:126) ) it follows H(cid:48)R(ϑ(cid:126)) =
AB C
R(ϑ(cid:126))H, from which follows in turn the well-known result that H(cid:48) is related to H through the
similarity transformation H(cid:48) = R(ϑ(cid:126))HR−1(ϑ(cid:126)). The invariance of the Hamiltonian under overall
rotations of the three particle system implies then
H = R(ϑ(cid:126))HR−1(ϑ(cid:126)) . (6.6)
For the following it is essential to note that H is not invariant under rotations of only (cid:126)r or ρ(cid:126) ,
AB C
but solely under simultaneous and identical rotations of (cid:126)r or ρ(cid:126) .
AB C
Following our description of rotations of single particle wave functions we express (6.5) according
to (5.48)
i i
R(ϑ(cid:126)) = exp(cid:18)− ϑ(cid:126)·J(cid:126)(1)(cid:19) exp(cid:18)− ϑ(cid:126)·J(cid:126)(2)(cid:19) (6.7)
(cid:126) (cid:126)
where the generators J(cid:126)(k) are differential operators acting on rˆ (k = 1) and on ρˆ (k = 2). For
AB C
example, according to (5.53, 5.55) holds
i ∂ ∂ i ∂ ∂
− J(1) = z − y ; − J(2) = ρ − ρ . (6.8)
(cid:126) 1 AB∂y AB∂z (cid:126) 3 y∂ρ x∂ρ
AB AB x y
Obviously, the commutation relationships
J(1), J(2) = 0 for p,q = 1,2,3 (6.9)
(cid:104) p q (cid:105)
hold since the components of J(cid:126)(k) are differential operators with respect to different variables. One
can equivalently express therefore (6.7)
i
R(ϑ(cid:126)) = exp(cid:18)− ϑ(cid:126)·(cid:126)J(cid:19) (6.10)
(cid:126)
144 Addition of Angular Momenta and Spin
where
(cid:126)J = J(cid:126)(1) + J(cid:126)(2) . (6.11)
By means of (6.11) we can write the condition (6.6) for rotational invariance of the Hamiltonian in
the form
i i
H = exp(cid:18)− ϑ(cid:126)·(cid:126)J(cid:19) H exp(cid:18)+ ϑ(cid:126)·(cid:126)J(cid:19) . (6.12)
(cid:126) (cid:126)
We consider this equation for infinitesimal rotations, i.e. for |ϑ(cid:126)| (cid:28) 1. To order O(|ϑ(cid:126)|) one obtains
i i i i
H ≈ (cid:18)11 − ϑ(cid:126)·(cid:126)J(cid:19) H (cid:18)11 + ϑ(cid:126)·(cid:126)J(cid:19) ≈ H + Hϑ(cid:126)·(cid:126)J − ϑ(cid:126)·(cid:126)JH . (6.13)
(cid:126) (cid:126) (cid:126) (cid:126)
Since this holds for any ϑ(cid:126) it follows H(cid:126)J − (cid:126)JH = 0 or, componentwise,
[H, J ] = 0 , k = 1,2,3 . (6.14)
k
We will refer in the following to J , k = 1,2,3 as the three components of the total angular
k
momentum operator.
The property (6.14) implies that the total angular momentum is conserved during the scattering
process, i.e., that energy, and the eigenvalues of(cid:126)J2 and J are good quantum numbers. To describe
3
the scattering process of AB + C most concisely one seeks eigenstates Y of(cid:126)J2 and J which can
JM 3
be expressed in terms of Y (rˆ )Y (ρˆ ).
(cid:96)1m1 AB (cid:96)2m2 C
Definition of Total Angular Momentum States
The commutation property (6.14) implies that the components of the total angular momentum
operator (6.12) each individually can have simultaneous eigenstates with the Hamiltonian. We
suspect, of course, that the components J , k = 1,2,3 cannot have simultaneous eigenstates a-
k
mong each other, a supposition which can be tested through the commutation properties of these
operators. One can show readily that the commutation relationships
[J ,J ] = i(cid:126)(cid:15) J (6.15)
k (cid:96) k(cid:96)m m
are satisfied, i.e., the operators J , k = 1,2,3 do not commute. For a proof one uses (6.9), the
k
properties [J(n),J(n)] = i(cid:126)(cid:15) J(n) for n = 1,2 together with the property [A,B + C] =
k (cid:96) k(cid:96)m m
[A,B] + [A,C].
We recognize, however, the important fact that the J obey the Lie algebra of SO(3). According
k
to the theorem above this property implies that one can construct eigenstates Y of J and of
JM 3
J2 = J2 + J2 + J2 (6.16)
1 2 3
following the procedure stated in the theorem above [c.f. Eqs. (5.71–5.81)]. In fact, we will find
that the states yield the basis B(cid:48) with the desired property of a maximal uncoupling of rotational
states.
Before we apply the procedure (5.71–5.81) we want to consider the relationship between Y and
JM
Y (rˆ )Y (ρˆ ). In the following we will use the notation
(cid:96)1m1 AB (cid:96)2m2 C
Ω = rˆ , Ω = ρˆ . (6.17)
1 AB 2 C
6.1: Clebsch-Gordan Coefficients 145
6.1 Clebsch-Gordan Coefficients
InordertodetermineY wenoticethatthestatesY (Ω )Y (Ω )arecharacterizedbyfour
JM (cid:96)1m1 1 (cid:96)2m2 2
quantum numbers corresponding to eigenvalues of J(1) 2, J(1), J(2) 2, and J(2). Since Y
(cid:2) (cid:3) 3 (cid:2) (cid:3) 3 JM
sofar specifies solely two quantum numbers, two further quantum numbers need to be specified
for a complete characterization of the total angular momentum states. The two missing quantum
numbers are (cid:96) and (cid:96) corresponding to the eigenvalues of J(1) 2 and J(2) 2. We, therefore,
1 2
(cid:2) (cid:3) (cid:2) (cid:3)
assume the expansion
Y ((cid:96) ,(cid:96) |Ω ,Ω ) = (JM|(cid:96) m (cid:96) m )Y (Ω )Y (Ω ) (6.18)
JM 1 2 1 2 (cid:88) 1 1 2 2 (cid:96)1m1 1 (cid:96)2m2 2
m1,m2
where the states Y ((cid:96) ,(cid:96) |Ω ,ρˆ ) are normalized. The expansion coefficients (JM|(cid:96) m (cid:96) m )
JM 1 2 1 C 1 1 2 2
are called the Clebsch-Gordan coefficients which we seek to determine now. These coefficients,
or the closely related Wigner 3j-coefficients introduced further below, play a cardinal role in the
mathematical description of microscopic physical systems. Equivalent coefficients exist for other
symmetry properties of multi–component systems, an important example being the symmetry
groupsSU(N)governingelementaryparticlesmadeupoftwoquarks,i.e.,mesons,andthreequarks,
i.e., baryons.
Exercise 6.1.1: ShowthatJ2,J , J(1) 2, J(2) 2, andJ(cid:126)(1)·J(cid:126)(2) commute. WhycanstatesY
3 JM
(cid:0) (cid:1) (cid:0) (cid:1)
then not be specified by 5 quantum numbers?
Properties of Clebsch-Gordan Coefficients
A few important properties of Clebsch-Gordan coefficients can be derived rather easily. We first
notice that Y in (6.18) is an eigenfunction of J , the eigenvalue being specified by the quantum
JM 3
number M, i.e.
J Y = (cid:126)M Y . (6.19)
3 JM JM
Noting J = J(1) + J(2) and applying this to the l.h.s. of (6.18) yields using the property
3 3 3
J(k)Y (Ω ) = (cid:126)m Y (Ω ), k = 1,2
3 (cid:96)kmk k k (cid:96)kmk k
M Y ((cid:96) ,(cid:96) |Ω ,Ω ) =
JM 1 2 1 2
(m + m )(JM|(cid:96) m (cid:96) m )Y (Ω )Y (Ω ) . (6.20)
(cid:88) 1 2 1 1 2 2 (cid:96)1m1 1 (cid:96)2m2 2
m1,m2
This equation can be satisfied only if the Clebsch-Gordan coefficients satisfy
(JM|(cid:96) m (cid:96) m ) = 0 for m + m (cid:54)= M . (6.21)
1 1 2 2 1 2
One can, hence, restrict the sum in (6.18) to avoid summation of vanishing terms
Y ((cid:96) ,(cid:96) |Ω ,Ω ) = (JM|(cid:96) m (cid:96) M −m )Y (Ω )Y (Ω ) . (6.22)
JM 1 2 1 2 (cid:88) 1 1 2 1 (cid:96)1m1 1 (cid:96)2m2 2
m1
We will not adopt such explicit summation since it leads to cumbersum notation. However, the
reader should always keep in mind that conditions equivalent to (6.21) hold.
146 Theory of Angular Momentum and Spin
The expansion (6.18) constitutes a change of an orthonormal basis
B((cid:96) ,(cid:96) ) = {Y (Ω )Y (Ω ),m = −(cid:96) ,−(cid:96) +1,...,(cid:96) ,
1 2 (cid:96)1m1 1 (cid:96)2m2 2 1 1 1 1
m = −(cid:96) ,−(cid:96) +1,...,(cid:96) } , (6.23)
2 2 2 2
(cid:48)
corresponding to the r.h.s., to a new basis B((cid:96) ,(cid:96) ) corresponding to the l.h.s. The orthonormality
1 2
property implies
(cid:90) dΩ1(cid:90) dΩ2 Y(cid:96)1m1(Ω1)Y(cid:96)2m2(Ω2)Y(cid:96)(cid:48)1m(cid:48)1Ω1)Y(cid:96)(cid:48)2m(cid:48)2(Ω2) = δ(cid:96)1(cid:96)(cid:48)1δm1m(cid:48)1δ(cid:96)2(cid:96)(cid:48)2δm2m(cid:48)2. (6.24)
The basis B((cid:96) ,(cid:96) ) has (2(cid:96) +1)(2(cid:96) +1) elements. The basis B(cid:48)((cid:96) ,(cid:96) ) is also orthonormal2 and
1 2 1 2 1 2
must have the same number of elements. For each quantum number J there should be 2J + 1
elements Y with M = −J,−J +1,...,J. However, it is not immediately obvious what the J–
JM
values are. Since Y represents the total angular momentum state and Y (Ω ) and Y (Ω )
JM (cid:96)1m1 1 (cid:96)2m2 2
theindividualangularmomentaonemaystartfromone’sclassicalnotionthatthesestatesrepresent
angularmomentumvectors J(cid:126), J(cid:126)(1) andJ(cid:126)(2), respectively. Inthis case the rangeof|J(cid:126)|–values would
be the interval [ |J(cid:126)(1)| − |J(cid:126)(2)| , |J(cid:126)(1)| + |J(cid:126)(2)|]. This obviously corresponds quantum mechanically
(cid:12) (cid:12)
(cid:12) (cid:12)
to a range of J–(cid:12)values J = |(cid:96)1(cid:12)−(cid:96)2|, |(cid:96)1−(cid:96)2|+1,...(cid:96)1+(cid:96)2. In fact, it holds
(cid:96)1+(cid:96)2
(2J + 1) = (2(cid:96) +1)(2(cid:96) +1) , (6.25)
(cid:88) 1 2
J=|(cid:96)1−(cid:96)2|
(cid:48)
i.e., the basis B((cid:96) ,(cid:96) ) should be
1 2
B = {Y ((cid:96) ,(cid:96) |Ω ,Ω ); J = |(cid:96) −(cid:96) |, |(cid:96) −(cid:96) |+1,(cid:96) +(cid:96) ,
2 JM 1 2 1 2 1 2 1 2 1 2
M = −J,−J +1,...,J} . (6.26)
We will show below in an explicit construction of the Clebsch-Gordan coefficients that, in fact, the
range of values assumed for J is correct. Our derivation below will also yield real values for the
Clebsch-Gordan coefficients.
Exercise 6.1.2: Prove Eq. (6.25)
We want to state now two summation conditions which follow from the orthonormality of the two
(cid:48)
basis sets B((cid:96) ,(cid:96) ) and B((cid:96) ,(cid:96) ). The property
1 2 1 2
(cid:90) dΩ1(cid:90) dΩ2 Y∗JM((cid:96)1,(cid:96)2|Ω1,Ω2)YJ(cid:48)M(cid:48)((cid:96)1,(cid:96)2|Ω1,Ω2) = δJJ(cid:48)δMM(cid:48) (6.27)
∗
together with (6.18) applied to YJM and to YJ(cid:48)M(cid:48) and with (6.24) yields
(cid:88) (JM|(cid:96)1m1(cid:96)2m2)∗(J(cid:48)M(cid:48)|(cid:96)1m1(cid:96)2m2) = δJJ(cid:48)δMM(cid:48) . (6.28)
m1,m2
2This property follows from the fact that the basis elements are eigenstates of hermitian operators with different
eigenvalues, and that the states can be normalized.
6.2: Construction of Clebsch-Gordan Coefficients 147
(cid:48)
The second summation condition starts from the fact that the basis sets B((cid:96) ,(cid:96) ) and B((cid:96) ,(cid:96) )
1 2 1 2
span the same function space. Hence, it is possible to expand Y (Ω )Y (Ω ) in terms of
(cid:96)1m1 1 (cid:96)2m2 2
Y ((cid:96) ,(cid:96) |Ω ,Ω ), i.e.,
JM 1 2 1 2
(cid:96)1+(cid:96)2 J
Y(cid:96)1m1(Ω1)Y(cid:96)2m2(Ω2) = (cid:88) (cid:88) cJ(cid:48)M(cid:48)YJ(cid:48)M(cid:48)((cid:96)1,(cid:96)2|Ω1,Ω2) , (6.29)
J(cid:48)=|(cid:96)1−(cid:96)2|M(cid:48)=−J
where the expansion coefficients are given by the respective scalar products in function space
cJ(cid:48)M(cid:48) = (cid:90) dΩ1(cid:90) dΩ2 Y∗J(cid:48)M(cid:48)((cid:96)1,(cid:96)2|Ω1,Ω2)Y(cid:96)1m1(Ω1)Y(cid:96)2m2(Ω2) . (6.30)
The latter property follows from multiplying (6.18) by Y∗ ((cid:96) ,(cid:96) |Ω ,Ω ) and integrating. The
J(cid:48)M(cid:48) 1 2 1 2
orthogonality property (6.27) yields
δJJ(cid:48)δMM(cid:48) = (cid:88) (JM|(cid:96)1m1(cid:96)2m2)cJ(cid:48)M(cid:48) . (6.31)
m1,m2
Comparisionwith(6.28)allowsonetoconcludethatthecoefficientscJ(cid:48)M(cid:48) areidenticalto(J(cid:48)M(cid:48)|(cid:96)1m1(cid:96)2m2)∗,
i.e.,
Y (Ω )Y (Ω )
(cid:96)1m1 1 (cid:96)2m2 2
(cid:96)1+(cid:96)2 J
= (cid:88) (cid:88) (J(cid:48)M(cid:48)|(cid:96)1m1(cid:96)2m2)∗YJ(cid:48)M(cid:48)((cid:96)1,(cid:96)2|Ω1,Ω2) , (6.32)
J(cid:48)=|(cid:96)1−(cid:96)2|M(cid:48)=−J
which complements (6.18). One can show readily using the same reasoning as applied in the
derivation of (6.28) from (6.18) that the Clebsch-Gordan coefficients obey the second summation
condition
(cid:88)(JM|(cid:96)1m1(cid:96)2m2)∗(JM|(cid:96)1m(cid:48)1(cid:96)2m(cid:48)2) = δm1m(cid:48)1δm2m(cid:48)2 . (6.33)
JM
The latter summation has not been restricted explicitly to allowed J–values, rather the convention
(JM|(cid:96) m (cid:96) m ) = 0 if J < |(cid:96) −(cid:96) |, or J > (cid:96) +(cid:96) (6.34)
1 1 2 2 1 2 1 2
has been assumed.
6.2 Construction of Clebsch-Gordan Coefficients
We will now construct the Clebsch-Gordan coefficients. The result of this construction will include
allthepropertiespreviewedabove. Atthispointweliketostressthattheconstructionwillbebased
onthetheorems(5.71–5.81)statedabove,i.e.,willbebasedsolelyonthecommutationpropertiesof
the operators(cid:126)J and J(cid:126)(k). We can, therefore, also apply the results, and actually also the properties
ofClebsch-Gordan coefficientsstatedabove,tocompositesystemsinvolvingspin-1 states. Asimilar
2
construction will also be applied to composite systems governed by other symmetry groups, e.g.,
the group SU(3) in case of meson multiplets involving two quarks, or baryons multiplets involving
three quarks.
148 Addition of Angular Momenta and Spin
For the construction of Y we will need the operators
JM
J± = J1 + iJ2 . (6.35)
The construction assumes a particular choice of J ∈ {|(cid:96) −(cid:96) |, |(cid:96) −(cid:96) |+1,...(cid:96) +(cid:96) } and for
1 2 1 2 1 2
such J–value seeks an expansion (6.18) which satisfies
J Y ((cid:96) ,(cid:96) |Ω ,Ω ) = 0 (6.36)
+ JJ 1 2 1 2
J Y ((cid:96) ,(cid:96) |Ω ,Ω ) = (cid:126)JY ((cid:96) ,(cid:96) |Ω ,Ω ) . (6.37)
3 JJ 1 2 1 2 JJ 1 2 1 2
The solution needs to be normalized. Having determined such Y we then construct the whole
JJ
family of functions X = {Y ((cid:96) ,(cid:96) |Ω ,Ω ), M = −J,−J +1,...J} by applying repeatedly
J JM 1 2 1 2
J−YJM+1((cid:96)1,(cid:96)2|Ω1,Ω2) = (cid:126) (J +M +1)(J −M)YJM((cid:96)1,(cid:96)2|Ω1,Ω2) . (6.38)
(cid:112)
for M = J −1,J −2,...,−J.
WeembarkonthesuggestedconstructionforthechoiceJ = (cid:96) +(cid:96) . Wefirstseekanunnormalized
1 2
solutionG andlaternormalize. TofindG westartfromtheobservationthatG representsthe
JJ JJ JJ
state with the largest possible quantum number J = (cid:96) + (cid:96) with the largest possible component
1 2
M = (cid:96) + (cid:96) along the z–axis. The corresponding classical total angular momentum vector(cid:126)J
1 2 class
would be obtained by aligning both J(cid:126)(1) and J(cid:126)(2) also along the z–axis and adding these two
class class
vectors. Quantum mechanically this corresponds to a state
G ((cid:96) ,(cid:96) |Ω ,Ω ) = Y (Ω )Y (Ω ) (6.39)
(cid:96)1+(cid:96)2,(cid:96)1+(cid:96)2 1 2 1 2 (cid:96)1(cid:96)1 1 (cid:96)2(cid:96)2 2
which we will try for a solution of (6.37). For this purpose we insert (6.39) into (6.37) and replace
according to (6.11) J by J(1) + J(2). We obtain using (5.66,5.68)
+ + +
J(1) + J(2) Y (Ω )Y (Ω ) (6.40)
(cid:16) + + (cid:17) (cid:96)1(cid:96)1 1 (cid:96)2(cid:96)2 2
= J(1)Y (Ω ) Y (Ω ) + Y (Ω ) J(2)Y (Ω ) = 0 .
(cid:16) + (cid:96)1(cid:96)1 1 (cid:17) (cid:96)2(cid:96)2 2 (cid:96)1(cid:96)1 1 (cid:16) + (cid:96)2(cid:96)2 2 (cid:17)
Similarly, we can demonstrate condition (6.25) using (6.11) and (5.64)
J(1) + J(2) Y (Ω )Y (Ω )
(cid:16) 3 3 (cid:17) (cid:96)1(cid:96)1 1 (cid:96)2(cid:96)2 2
= J(1)Y (Ω ) Y (Ω ) + Y (Ω ) J(2)Y (Ω )
(cid:16) 3 (cid:96)1(cid:96)1 1 (cid:17) (cid:96)2(cid:96)2 2 (cid:96)1(cid:96)1 1 (cid:16) 3 (cid:96)2(cid:96)2 2 (cid:17)
= (cid:126)((cid:96) + (cid:96) )Y (Ω )Y (Ω ) . (6.41)
1 2 (cid:96)1(cid:96)1 1 (cid:96)2(cid:96)2 2
In fact, we can also demonstrate using (??) that G ((cid:96) ,(cid:96) |Ω ,Ω ) is normalized
(cid:96)1+(cid:96)2,(cid:96)1+(cid:96)2 1 2 1 2
(cid:90) dΩ1 (cid:90) dΩ2G(cid:96)1+(cid:96)2,(cid:96)1+(cid:96)2((cid:96)1,(cid:96)2|Ω1,Ω2)
= (cid:18)(cid:90) dΩ1Y(cid:96)1(cid:96)1(Ω1)(cid:19) (cid:18)(cid:90) dΩ2Y(cid:96)2(cid:96)2(Ω2)(cid:19) = 1 . (6.42)
We, therefore, have shown
Y ((cid:96) ,(cid:96) |Ω ,Ω ) = Y (Ω )Y (Ω ) . (6.43)
(cid:96)1+(cid:96)2,(cid:96)1+(cid:96)2 1 2 1 2 (cid:96)1(cid:96)1 1 (cid:96)2(cid:96)2 2
6.2: Construction of Clebsch-Gordan Coefficients 149
Wenowemployproperty(6.38)toconstructthefamilyoffunctionsB = {Y ((cid:96) ,(cid:96) |(cid:10) ,(cid:10) ), M =
(cid:96)(cid:49)+(cid:96)(cid:50) (cid:96)(cid:49)+(cid:96)(cid:50)M (cid:49) (cid:50) (cid:49) (cid:50)
−((cid:96) +(cid:96) ),...,((cid:96) +(cid:96) )}. We demonstrate the procedure explicitly only for M = (cid:96) +(cid:96) −1.
(cid:49) (cid:50) (cid:49) (cid:50) √ 1 2
T√he r.h.s. of (6.38) yields with J− = J−(1) + J−(1) the expression (cid:126) 2(cid:96)1Y(cid:96)1(cid:96)1−1(Ω1)Y(cid:96)2(cid:96)2(Ω2) +
(cid:126) 2(cid:96)2Y(cid:96)1(cid:96)1−1(Ω1)Y(cid:96)2(cid:96)2−1(Ω2). The l.h.s. of (6.38) yields (cid:126)(cid:112)2((cid:96)1+(cid:96)2)Y(cid:96)1+(cid:96)2(cid:96)1+(cid:96)2−1((cid:96)1,(cid:96)2|Ω1,Ω2).
One obtains then
Y(cid:96)1+(cid:96)2(cid:96)1+(cid:96)2−1((cid:96)1,(cid:96)2|Ω1,Ω2) = (6.44)
(cid:113)(cid:96)1(cid:96)+1(cid:96)2 Y(cid:96)1(cid:96)1−1(Ω1)Y(cid:96)2(cid:96)2(Ω2) +(cid:113)(cid:96)1(cid:96)+2(cid:96)2 Y(cid:96)1(cid:96)1(Ω1)Y(cid:96)2(cid:96)2−1(Ω2) .
This construction can be continued to obtain all 2((cid:96) +(cid:96) ) + 1 elements of B and, thereby,
1 2 (cid:96)(cid:49)+(cid:96)(cid:50)
all the Clebsch-Gordan coefficients ((cid:96) +(cid:96) M|(cid:96) m (cid:96) m ). We have provided in Table 1 the explicit
1 2 1 1 2 2
form of Y ((cid:96) (cid:96) |Ω Ω ) for (cid:96) = 2 and (cid:96) = 1 to illustrate the construction. The reader should
JM 1 2 1 2 1 2
familiarize himself with the entries of the Table, in particular, with the symmetry pattern and with
the terms Y Y contributing to each Y .
(cid:96)1m1 (cid:96)2m2 JM
WeliketoconstructnowthefamilyoftotalangularmomentumfunctionsB(cid:96)(cid:49)+(cid:96)(cid:50)−(cid:49) = {Y(cid:96)(cid:49)+(cid:96)(cid:50)−(cid:49)M((cid:96)(cid:49),(cid:96)(cid:50)|(cid:10)(cid:49),(cid:10)(cid:50)), M =
−((cid:96) + (cid:96) − (cid:49)),...,((cid:96) + (cid:96) − (cid:49))}. We seek for this purpose first an unnormalized solution
(cid:49) (cid:50) (cid:49) (cid:50)
G(cid:96)1+(cid:96)2−1(cid:96)1+(cid:96)2−1 of (6.36, 6.37). According to the condition (6.21) we set
G(cid:96)1+(cid:96)2−1(cid:96)1+(cid:96)2−1((cid:96)1(cid:96)2|Ω1Ω2) = Y(cid:96)1(cid:96)1−1(Ω1)Y(cid:96)2(cid:96)2(Ω2) + cY(cid:96)1(cid:96)1(Ω1)Y(cid:96)2(cid:96)2−1(Ω2) (6.45)
for some unknown constant c. One can readily show that (6.37) is satisfied. To demonstrate (6.36)
we proceed as above and obtain
(cid:16)J+(1)Y(cid:96)1(cid:96)1−1(Ω1)(cid:17)Y(cid:96)2(cid:96)2(Ω2) + cY(cid:96)1(cid:96)1(Ω1)(cid:16)J+(2)Y(cid:96)2(cid:96)2−1(Ω2)(cid:17)
= 2(cid:96) + c 2(cid:96) Y (Ω )Y (Ω ) = 0 . (6.46)
(cid:16)(cid:112) 1 (cid:112) 2(cid:17) (cid:96)1(cid:96)1 1 (cid:96)2(cid:96)2 2
Tosatisfythisequationoneneedstochoosec = −(cid:112)(cid:96)1/(cid:96)2. WehavetherebydeterminedG(cid:96)1+(cid:96)2−1(cid:96)1+(cid:96)2−1
in (6.45). Normalization yields furthermore
Y(cid:96)1+(cid:96)2−1(cid:96)1+(cid:96)2−1((cid:96)1(cid:96)2|Ω1Ω2) (6.47)
(cid:96) (cid:96)
= (cid:114)(cid:96) +2(cid:96) Y(cid:96)1(cid:96)1−1(Ω1)Y(cid:96)2(cid:96)2(Ω2) − (cid:114)(cid:96) +1(cid:96) Y(cid:96)1(cid:96)1(Ω1)Y(cid:96)2(cid:96)2−1(Ω2) .
1 2 1 2
This expression is orthogonal to (6.39) as required by (6.27).
Expression (6.47) can serve now to obtain recursively the elements of the family B(cid:96)(cid:49)+(cid:96)(cid:50)−(cid:49) for
M = (cid:96) +(cid:96) −2,(cid:96) +(cid:96) −3,...,−((cid:96) +(cid:96) −1). Having constructed this family we have determined
1 2 1 2 1 2
the Clebsch-Gordan coefficients ((cid:96) + (cid:96) − 1M|(cid:96) m (cid:96) m ). The result is illustrated for the case
1 2 1 1 2 2
(cid:96) = 2, (cid:96) = 1 in Table 1.
1 2
One can obviously continue the construction outlined to determine Y(cid:96)1+(cid:96)2−2(cid:96)1+(cid:96)2−2, etc. and all
total angular momentum functions for a given choice of (cid:96) and (cid:96) .
1 2
Exercise 6.2.1: Construct following the procedure above the three functions Y ((cid:96) ,(cid:96) |Ω ,Ω2)
JM 1 2 1
for M = (cid:96) +(cid:96) − 2 and J = (cid:96) +(cid:96) ,(cid:96) +(cid:96) − 1,(cid:96) +(cid:96) − 2. Show that the resulting functions
1 2 1 2 1 2 1 2
are orthonormal.
150 Addition of Angular Momenta and Spin
Y (Ω )Y (Ω )
22 1 11 2
Y (2,1|Ω ,Ω2) 1
33 1
Y (Ω )Y (Ω ) Y (Ω )Y (Ω )
21 1 11 2 22 1 10 2
Y (2,1|Ω ,Ω2) 2 (cid:39) 0.816497 1 (cid:39) 0.57735
32 1 (cid:113)3 (cid:113)3
Y (2,1|Ω ,Ω2) − 1 (cid:39) −0.57735 2 (cid:39) 0.816497
22 1 (cid:113)3 (cid:113)3
Y20(Ω1)Y11(Ω2) Y21(Ω1)Y10(Ω2) Y22(Ω1)Y1−1(Ω2)
Y (2,1|Ω ,Ω2) 2 (cid:39) 0.632456 8 (cid:39) 0.730297 1 (cid:39) 0.258199
31 1 (cid:113)5 (cid:113)15 (cid:113)15
Y (2,1|Ω ,Ω2) − 1 (cid:39) −0.707107 1 (cid:39) 0.408248 1 (cid:39) 0.57735
21 1 (cid:113)2 (cid:113)6 (cid:113)3
Y (2,1|Ω ,Ω2) 1 (cid:39) 0.316228 − 3 (cid:39) −0.547723 3 (cid:39) 0.774597
11 1 (cid:113)10 (cid:113)10 (cid:113)5
Y2−1(Ω1)Y11(Ω2) Y20(Ω1)Y10(Ω2) Y21(Ω1)Y1−1(Ω2)
Y (2,1|Ω ,Ω2) 1 (cid:39) 0.447214 3 (cid:39) 0.774597 1 (cid:39) 0.447214
30 1 (cid:113)5 (cid:113)5 (cid:113)5
Y (2,1|Ω ,Ω2) − 1 (cid:39) −0.707107 0 1 (cid:39) 0.707107
20 1 (cid:113)2 (cid:113)2
Y (2,1|Ω ,Ω2) 3 (cid:39) 0.547723 − 2 (cid:39) −0.632456 3 (cid:39) 0.547723
10 1 (cid:113)10 (cid:113)5 (cid:113)10
Y2−2(Ω1)Y11(Ω2) Y2−1(Ω1)Y10(Ω2) Y20(Ω1)Y1−1(Ω2)
Y3−1(2,1|Ω1,Ω2) (cid:113)115 (cid:39) 0.258199 (cid:113)185 (cid:39) 0.730297 (cid:113)25 (cid:39) 0.632456
Y2−1(2,1|Ω1,Ω2) −(cid:113)13 (cid:39) −0.57735 −(cid:113)16 (cid:39) −0.408248 (cid:113)12 (cid:39) 0.707107
Y1−1(2,1|Ω1,Ω2) (cid:113)35 (cid:39) 0.774597 −(cid:113)130 (cid:39) −0.547723 (cid:113)110 (cid:39) 0.316228
Y2−2(Ω1)Y10(Ω2) Y2−1(Ω1)Y1−1(Ω2)
Y3−2(2,1|Ω1,Ω2) (cid:113)13 (cid:39) 0.57735 (cid:113)23 (cid:39) 0.816497
Y2−2(2,1|Ω1,Ω2) −(cid:113)23 (cid:39) −0.816497 (cid:113)13 (cid:39) 0.57735
Y2−2(Ω1)Y1−1(Ω2)
Y3−3(2,1|Ω1,Ω2) 1
Table 6.1: Some explicit analytical and numerical values of Clebsch-Gordan coefficients and their
relationship to the total angular momentum wave functions and single particle angular momentum
wave functions.