Download Chapter 2 Theory of angular momentum

Chapter 2
Theory of angular momentum
2.1
2.1.1
Importance of angular momentum
Orbital angular momentum in Quantum Mechanics
As seen in the classical mechanics course, angular momentum plays an important role
in the description of the motion of physical systems; the total angular momentum of an
isolated physical system is a constant of motion.
All the properties of the classical angular momentum have their counterparts in quantum
~ with three
mechanics. Through the correspondence principle, we associate an operator L
~2
components Lx , Ly , Lz to the observable angular momentum . In a central potential, L
and Lz commute with the Hamiltonian.
Classically, the angular momentum of a particle (with respect to the origin) is given by:
~ = ~r × p~,
L
Lx = ypz − zpy ,
Ly = zpx − xpz ,
Lz = xpy − ypx .
The corresponding quantum operators are obtained by the standard prescription:
px → −i~
∂
,
∂x
py → −i~
∂
,
∂y
∂
.
∂z
And, from the canonical commutation relations:
pz → −i~
[ri , pj ] = i~δij ,
33
34
CHAPTER 2. THEORY OF ANGULAR MOMENTUM
~ 2 , Li :
we can derive the commutation relations for the components of L
[Lx , Ly ] = [ypz − zpy , zpx − xpz ]
= [ypz , zpx ] − [ypz , xpz ] − [zpy , zpx ] +[zpy , xpz ]
| {z } | {z }
0
0
= y[pz , z]px + x[z, pz ]py
= −i~ypx + i~xpy = i~Lz ;
→ [Li , Lj ] = i~ǫijk Lk .
For a system of N particles (with spin 0),
~ =
L
N
X
~ n,
L
~ n = ~rn × p~n .
where L
n=1
2.1.2
Spin
• In classical mechanics, a rigid object admits two kinds of angular momentum:
~ = ~r × ~p), associated with the motion of the center of mass;
– orbital (L
~ = Iω), associated with the motion about the center of mass.
– spin (|S|
F.i., the Earth has an orbital angular momentum attributable to its annual revolution around the Sun, and a spin angular momentum coming from its daily rotation
about the north-south axis. However, this description is only a matter of conve~ can be obtained by adding the angular momenta, L
~ n , of all the
nience since S
particles constituting the Earth.
• In quantum mechanics, there is a fundamental difference. A quantum particle,
~ (which is
f.i., an electron moving around the nucleus, has an orbital momentum L
quantized as we saw in QM I) associated with the motion around the nucleus as
~ which has nothing to do with motion in
well as an intrinsic angular momentum S,
space. An electron is an elementary particle and its spin angular momentum cannot
be decomposed into orbital angular momenta of constituting parts.
Intrinsic angular momentum of a particle:
~ S = 2 µB S.
~
M
~
Quantum description of spin by Pauli (valid in the non-relativistic limit)
Postulates:
~ is an angular momentum operator, which means that its 3
1. the spin operator S
components are observables that satisfy the commutation relations:
[Si , Sj ] = i~ǫijk Sk ;
2.2. GENERALIZATION: ANGULAR MOMENTUM J~
35
~ 2 and Sz
2. the spin operators act in a new space, the “spin state space”, Es , where S
constitute a complete set of commuting observables, c.s.c.o. . The space Es is thus
~ 2 and Sz :
spanned by the set of eigenstates |s, mi common to S
~ 2 |s, mi = s(s + 1)~2 |s, mi ,
S
Sz |s, mi = m~ |s, mi .
According to the general theory of angular momentum, s must be integral or halfintegral, and m takes the values: m = −s, −s + 1, . . . , s (we will see this in the
next section). A given particle is defined by a unique value of s. The spin state
~ 2 with
space Es is of finite dimension 2s + 1 and all spin states are eigenvectors of S
the eigenvalue s(s + 1)~2 .
3. The state space E of the particle being considered is the tensor product of Er and
Es :
E = Er ⊗ Es .
Correspondingly, all spin abservables commute with all orbital observables. Except
for the particular case of s = 0, it is not sufficient to specify a ket of Er (i.e., a
square integrable wave function) to characterize a state of the particle. This means
that the observables x̂, ŷ, ẑ (or p̂x , p̂y , p̂z ) do not constitute a c.s.c.o. in the space
E of the particle. It is also necessary to know the spin state of the particle.
2.2
Generalization: angular momentum J~
The three operators associated with the components of an arbitrary classical angular
momentum satisfy the commutation relations:
[Li , Lj ] = i~ǫijk Lk .
The origin of this relation lies in the geometric properties of rotations in 3D space.
Let us adopt a more general point of view and define an angular momentum J~ as a set of
three observables Jx , Jy , Jz which satisfies:
[Ji , Jj ] = i~ǫijk Jk ,
J~ 2 = Jx2 + Jy2 + Jz2 ,
[J~ 2 , Jz ] = 0.
Statement of the problem: since the components of an arbitrary angular momentum J~ do
not commute, they are not simultaneously diagonalizable.
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CHAPTER 2. THEORY OF ANGULAR MOMENTUM
We shall therefore seek the system of eigenvectors, φ, common to J~ 2 and Jz :
J~ 2 φ = λφ,
Jz φ = µφ.
The J+ and J− operators (Ladder operators)
The technique that we now consider is very similar to the one we applied for the harmonic
oscillator.
We define operators J+ and J− as:
J± ≡ Jx ± iJy .
Properties of J+ and J− :
1. J+ and J− do not commute with Jz :
[Jz , J± ] = [Jz , Jx ] ± i[Jz , Jy ] = i~Jy ± i(−i~Jx )
= ±~(Jx ± iJy ),
→
2. J+ and J− commute with J~ 2 :
[Jz , J± ] = ±~J± ,
[J~ 2 , J± ] = 0;
3. if φ is an eigenfunction of J~ 2 and Jz , (J± φ) is also an eigenfunction of these operators:
J~ 2 (J± φ) = J± (J~ 2 φ) = J± (λφ) = λ(J± )φ;
(J± φ) is an eigenfunction of J~ 2 with the same eigenvalue as φ.
Using [Jz , J± ] = ±~J± , we will prove that J± φ is an eigenfunction of Jz :
Jz (J± φ) = (Jz J± − J± Jz )φ + J± Jz φ
= ±~J± φ + J± (µφ)
= (µ ± ~)J± φ,
J+ φ: an eigenfunction of Jz with the eigenvalue µ + ~;
J+ : raising operator (it raises the eigenvalue of Jz by ~);
J− φ: an eigenfunction of Jz with the eigenvalue µ − ~;
J− : lowering operator.
2.2. GENERALIZATION: ANGULAR MOMENTUM J~
37
For a given value of λ (J~ 2 φ = λφ), we obtain a
”ladder” of states with each ”rung” separated from its
neighbors by one unit of ~ in the eigenvalue of Jz (Fig.
2.1)
The process of “climbing the ladder” can not go forever.
Eventually, we are going to reach a state for which the zcomponent of J~ exceeds its total value, and this can not be.
Therefore, there must be a top rung, φt , for which:
Figure 2.1: The ladder of Jz
eigenstates
J+ φt = 0.
Let ~j be the eigenvalue of Jz at this top rung:
Jz φt = ~jφt .
Let us now express J~ 2 in terms of J+ and J− . From
J+ J− = (Jx + iJy )(Jx − iJy ) = Jx2 + Jy2 − i(Jx Jy − Jy Jx )
= J~ 2 − Jz2 − i(i~Jz ),
we have:
and
J~ 2 = J+ J− + Jz2 − ~Jz
J~ 2 = J− J+ + Jz2 + ~Jz .
Then,
J~ 2 φt = (J− J+ + Jz2 + ~Jz )φt = (0 + ~2 j 2 + ~2 j)φt
= ~2 j(j + 1)φt .
Comparing this result with J~ 2 φt = λφt , λ is therefore related to j as:
λ = ~2 j(j + 1).
(2.1)
The same follows for the bottom rung:
J− φb = 0,
Jz φb = ~j̃φb ,
J~ 2 φb = (J+ J− + Jz2 − ~Jz )φb = (0 + ~2 j̃ − ~2 j̃)φb
= ~2 j̃(j̃ − 1)φb ,
and
λ = ~2 j̃(j̃ − 1).
(2.2)
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CHAPTER 2. THEORY OF ANGULAR MOMENTUM
Comparing eq. (2.1) and (2.2),
j(j + 1) = j̃(j̃ − 1)
→
j̃ = −j.
j̃ = −j means that the eigenvalues of Jz are m~, where m goes from −j to +j in N
integer steps. In particular, j = −j + N and hence j = N/2, so j must be an integer or
half-integer.
The eigenfunctions of J~ 2 and Jz acquire now corresponding labels:
2
m
J~ 2 φm
j = ~ j(j + 1)φj ,
m
Jz φ m
j = ~mφj ,
where
1
3
j = 0, , 1, , . . . ,
m = −j, −j + 1, . . . , j − 1, j.
2
2
For a given value of j, there are 2j + 1 different values of m (2j + 1 ”rungs” on the ladder).
~ we saw in QMI that the eigenfunctions of L
~ 2 and Lz were the spherical
Note: for J~ = L,
harmonics Ylm . We showed that in this case only l integer had physical meaning.
2.2.1
Special properties of J = S = 1/2
In the case of S = 1/2, the spin space Es is two-dimensional. We take as a basis the
orthonormal system of eigenkets
{|+i , |−i}
~ 2 and Sz , which satisfies the equations:
common to S
~ 2 |±i = 3 ~2 |±i ,
S
4
1
Sz |±i = ± ~ |±i ,
2
h+ |−i = 0,
h+ |+i = h− |−i = 1,
|+i h+| + |−i h−| = 1.
The most general spin state χ can be constructed out of |+i and |−i as:
|χi = c+ |+i + c− |−i ,
with c+ , c− ∈ C.
~ 2 with eigenvalue 3~2 /4.
All the kets of Es are eigenfunctions of S
The eigenvalues of S+ and S− ,
S± = Sx ± iSy ,
are:
S+ |+i = 0,
S+ |−i = ~ |+i ,
2.2. GENERALIZATION: ANGULAR MOMENTUM J~
S− |−i = 0,
39
S− |+i = ~ |−i .
Any operator acting in Es can be represented in the {|+i , |−i} basis by 2 × 2 matrices,
σx , σy , σz , called Pauli matrices:
~ = ~ ~σ,
S
2
1 0
0 −i
0 1
.
,
σx =
,
σy =
σx =
0 −1
i 0
1 0
Some of the properties of ~σ and its components are:
σi2 = 1,
[σi , σj ] = 2iσk ,
Tr σi = 0,
Det σi = −1,
~
~
~
~
~
~
~ and B
~ are arbitrary vectors).
~σ · A ~σ · B = A · B + iσ A × B
(A
2.2.2
Non-relativistic description of a spin-1/2 particle
We know how to describe separately the external (orbital) and the internal (spin) degrees
of freedom. But how do we represent their combination?
The quantum state of an electron is characterized by a ket belonging to the space E,
which is a tensor product of Er and Es , following the third Pauli postulate. We can
consider various c.s.c.o. in E, f.i.,
~ 2 , Sz },
{x, y, z, S
~ 2 , Sz },
{px , py , pz , S
or
~ 2 , Lz , S
~ 2 , Sz }.
{H, L
Let us consider the first set. We take as a basis of E the ket of vectors obtained from the
tensor product of the kets |~ri ≡ |x, y, zi of Er and the kets |χi of Es :
|~r, χi ≡ |x, y, z, χi = |~ri ⊗ |χi .
|~r, χi are the eigenvectors of x, y, z, S 2 and Sz :
x̂ |~r, χi = x |~r, χi ,
ŷ |~r, χi = y |~r, χi ,
ẑ |~r, χi = z |~r, χi ,
3
Ŝ 2 |~r, χi = ~2 |~r, χi ,
4
40
CHAPTER 2. THEORY OF ANGULAR MOMENTUM
~
Sz |~r, χi = χ |~r, χi ,
2
′
′
h~r , χ |~r, χi = δχ′ χ δ(~r ′ − ~r),
Z
Z
XZ
3
3
d r |~r, χi h~r, χ| = d r |~r, +i h~r, +| + d3 r |~r, −i h~r, −| = 1.
χ
{|~r, χi} representation: SPINORS
Any state |Ψi of the space E can be expanded in the {|~r, χi} basis:
XZ
|Ψi =
d3 r |~r, χi h~r, χ |Ψi
χ
and
hΨ| =
=
XZ
d3 r hΨ |~r, χi h~r, χ|
χ
XZ
d3 r Ψ∗ (~r, χ) h~r, χ| ,
χ
with
h~r, χ |Ψi = Ψχ (~r)
→
The two-component spinor :
[Ψ](~r) =
and
[Ψ]† (~r) =
The scalar product:
Ψ+ (~r) / Ψ−(~r).
Ψ+ (~r)
Ψ− (~r)
Ψ∗+ (~r) Ψ∗− (~r)
Z
.
d3 r [Ψ]† (~r)[φ](~r),
Z
Z
3
†
hΨ |Ψi = d r [Ψ] (~r)[Ψ](~r) = d3 r [|Ψ+ (~r)|2 + |Ψ− (~r)|2 ] = 1.
hΨ |φi =
If the state vector under consideration is of the type:
|Ψi = |ϕi ⊗ |χi ,
R
with |ϕi = d3 r ϕ(~r) |~ri ∈ Er ,
|χi = c+ |+i + c− |−i ∈ Es ,
the spinor associated with it takes the form:
ϕ(~r)c+
c+
[Ψ](~r) =
= ϕ(~r)
ϕ(~r)c−
c−
and
2
hΨ |Ψi = hϕ |ϕi hχ |χi = |c+ | + |c− |
2
Z
d3 r |ϕ(~r)|2.
41
2.3. ANGULAR MOMENTUM AND ROTATIONS
Mixed operators:
2
~ · P~ ] = ~ (σx px + σy py + σz pz ) = ~
[S
2
2i
2.3
∂
∂x
∂
∂z
∂
+ i ∂y
∂
∂x
∂
− i ∂y
∂
− ∂z
.
Angular momentum and rotations
We mentioned that the commutation relations between the components of an angular
momentum are the expression of the geometrical properties of rotations in ordinary 3D
space. Here we want to show that and investigate the relation between rotation and angular momentum operators.
Consider a state described by |Ψi at time t. We perform a rotation R in this system,
after which its new quantum state is determined by |Ψ′ i. We define R̂ as a linear operator
acting in the state space E associated with the geometrical rotation R such that:
|Ψ′ i = R̂ |Ψi ,
In order to understand the relation between the geometrical rotation R, which operates
in ordinary space, and its image R̂, which acts in the state space, we want to proceed as
follows:
• (i) review the main properties of geometrical rotations R;
• (ii) consider the example of a spinless particle to define the rotation operator R̂;
~ and show that the commutation relations be• (iii) determine R̂ as a function of L
tween Li correspond to geometrical characteristics of the rotation R;
• (iv) classify observables according to how they transform under a rotation (scalar,
vector, tensor);
• (v) discuss the importance of rotation invariance.
2.3.1
Definition of rotation
A rotation R is a one-to-one transformation of the three dimensional space which conserves:
1. a point in space;
2. the angles;
3. the distances;
4. handeness of the reference frame.
42
CHAPTER 2. THEORY OF ANGULAR MOMENTUM
A rotation can be characterized by the axis of rotation (unit vector ~u) and the angle of rotation α:
α
~ = α~u,
α(0 ≤ α ≤ 2π).
Thus, to determine a rotation, three parameters are required.
To characterize a rotation, one can also choose the Euler angles.
We will use R~u (α) to denote a geometrical rotation through an
angle α about the axis defined by the unit vector ~u.
The set of rotations R constitutes a group since they have the following properties:
• the product of two rotations is a rotation;
• there exists an identity rotation;
• ∀ R~u (α) ∃ R−~u (α), the inverse rotation;
• R~u (α)R~u′ (α′ ) 6= R~u′ (α′ )R~u (α);
• but R~u (α)R~u (α′ ) = R~u (α + α′) = R~u (α′ )R~u (α).
2.3.2
Orthogonal group
A rotation can be specified by:
• the axis of rotation and the angle of rotation or
• by a 3 × 3 orthogonal matrix R,
RRT = 1,
since the effect of successive rotations can be obtained by multiplying the appropriate orthogonal matrices;
The set of 3 × 3 orthogonal matrices forms a group (as it should be if they represent
rotations), which is called
SO(3): S stands for special,
O stands for orthogonal,
3 stands for three dimensions.
Since only rotation operations are included, we have SO(3) rather than O(3) (which can
include the inversion operation also).
2.3.3
Infinitesimal rotations
2.3. ANGULAR MOMENTUM AND ROTATIONS
43
An infinitesimal rotation,
R~u (dα),
is defined as a rotation which is infinitesimally close to the
identity rotation. The transformation of a vector ~v under
an infinitesimal rotation can be written, to first order in dα
as:
R~u (dα)~v = ~v + dα ~u × ~v .
Since the angle of rotation can vary continuously, every finite
rotation can be decomposed into an infinite number of infinitesimal rotations as:
R~u (α + dα) = R~u (α)R~u (dα) = R~u (dα)R~u (α).
Thus, the study of the rotation group can be reduced to the examination of infinitesimal
rotations.
2.3.4
Rotation operators in state space
In order to introduce the rotation operator, we consider a physical system composed of a
single spinless particle in 3D space:
|Ψi ∈ Er
and
Ψ(~r) = h~r |Ψi .
We perform a rotation R on this system which associates the point ~r0 (x0 , y0 , z0 ) of space
with the point ~r ′0 (x′0 , y0′ , z0′ ) in such a way that:
~r ′0 = R~r0 .
Let |Ψ′ i be the state vector of the system after rotation and Ψ′ (~r) = h~r |Ψ′ i be the
corresponding wavefunction. It should be that
Ψ′ (~r ′0 ) = Ψ(~r0 ),
i.e., the value of the initial wave function Ψ(~r) at the point ~r0 should be after rotation
the value of the final wavefunction Ψ′ (~r) at the point ~r ′0 given by Ψ′ (~r ′0 ). Then,
Ψ′ (~r ′0 ) = Ψ(R−1~r ′0 ).
Since this equation is valid for any point ~r ′0 in space, we write it as:
Ψ′ (~r) = Ψ(R−1~r).
We define the operator R̂:
Then, eq. (2.3) becomes:
|Ψ′ i = R̂ |Ψi .
(2.3)
44
CHAPTER 2. THEORY OF ANGULAR MOMENTUM
h~r| R̂ |Ψi = hR−1~r |Ψi .
|R−1~ri is the basis ket of this representation determined by the components of the vector
R−1~r.
Properties of R̂:
• R̂ is linear:
Also,
|Ψi = λ1 |Ψ1 i + λ2 |Ψ2 i .
h~r| R̂ |Ψi = λ1 R−1~r |Ψ1 i + λ2 R−1~r |Ψ2 i
= λ1 h~r| R̂ |Ψ1 i + λ2 h~r| R̂ |Ψ2 i
and
R̂ |Ψi = R̂ [λ1 |Ψ1 i + λ2 |Ψ2 i] = λ1 R̂ |Ψ1 i + λ2 R̂ |Ψ2 i .
• R̂ is unitary: the unitarity of R̂,
is proven by considering
and its hermitian conjugate,
R̂R̂† = R̂† R̂ = 1,
h~r| R̂ = R−1~r
R̂† |~ri = R−1~r ,
and recalling that the ket |~ri represents a state in which the particle is perfectly
localized at |~ri:
R̂ |~ri = |R~ri .
Because of its unitarity, R̂ conserves the scalar product and the norm of the vectors
that it transforms:
|Ψ′ i = R̂ |Ψi
→ hφ′ |Ψ′ i = hφ |Ψi .
′
|φ i = R̂ |φi
• The set of operators R̂ constitutes a representation of the rotation group.
Indeed, since the product of two geometrical rotations is always a rotation:
R2 R1 = R3 ,
out of the definition of the operator R̂ it follows that
R̂2 R̂1 = R̂3 .
This means that the correspondence
R → R̂
conserves the group law.
45
2.3. ANGULAR MOMENTUM AND ROTATIONS
2.3.5
Rotation operators in terms of angular momentum observables
We consider an infinitesimal rotation about the Oz axis, Rêz (dα). If we apply it to
a particle whose state is described by the wavefunction Ψ(~r), the wavefunction Ψ′ (~r)
associated with the state of the particle after rotation is:
(dα)~r].
Ψ′ (~r) = Ψ[Rê−1
z
Since

 x + y dα
−1
y − x dα
Rêz (dα)~r = R−êz (dα)~r = ~r − dα êz × ~r

z
Ψ′ (x, y, z) = Ψ(x + y dα, y − x dα, z),
→
which yields, to first order in dα:
∂Ψ
∂Ψ
Ψ (x, y, z) = Ψ(x, y, z) + dα y
−x
∂x
∂y
∂
∂
Ψ
−y
= Ψ(x, y, z) − dα x
∂y
∂x
′
Therefore,
→
∂
i
∂
x
= L̂z .
−y
∂y
∂x
~
i
Ψ (~r) = h~r |Ψ i = h~r| 1 − dα L̂z |Ψi .
~
′
′
Since, by definition,
|Ψ′ i = R̂êz (dα) |Ψi ,
→
R̂êz (dα) = 1 −
In general,
R~u (dα) = 1 −
i
dα L̂z
~
1
~ · ~u.
dα L
~
Finite rotation
R̂êz (α + dα) = R̂êz (α)R̂êz (dα),
i
R̂êz (α + dα) = R̂êz (α) 1 − dα L̂z ,
~
from which follows that
i
R̂ê (α + dα) − R̂êz (α) = − dα R̂êz (α)L̂z .
~
R̂~ez and L̂z commute:
46
CHAPTER 2. THEORY OF ANGULAR MOMENTUM
R̂êz (α) = e−i/~ αLz .
In the case of an arbitrary unitary vector ~u:
~
R~u (α) = e−i/~ αL·~u .
~ are the generators of the infinitesimal rotation operators and define the Lie algebra of
L
the rotation group. Important:
R̂~u (α) = e−i/~ α(Lx ux +Ly uy +Lz uz ) 6= e−i/~ αLx ux e−i/~ αLy uy e−i/~ αLz uz ,
~
[R̂~u (α)]† = ei/~ αL·~u
→
~ is hermitian
R̂ is unitary since L
R̂~u (2π) = 1.
2.3.6
System of several spinless particles
Let us consider a system of two spinless particles (1) and (2). The state space of such a
system is the tensor product of the state spaces Er1 and Er2 :
E = E r1 ⊗ E r2 .
The state of the system, |Ψi, is formed by the particle (1) in the state |φ(1)i and the
particle (2) in the state |φ(2)i:
|Ψi = |φ(1)i ⊗ |φ(2)i .
We perform a rotation through an angle α about ~u on this two-particle system. The state
of the system after the rotation is:
h
i h
i
|Ψ′ i = |φ′ (1)i ⊗ |φ′ (2)i = R̂~u1 (α) |φ(1)i ⊗ R̂~u2 (α) |φ(2)i ,
(2.4)
with
~
~ 1 = ~r1 × p~1 )
(L
~
~ 2 = ~r2 × ~p2 ).
(L
R~u1 (α) = e−i/~ αL1 ·~u
and
R~u2 (α) = e−i/~ αL2 ·~u
By considering the definition of the tensor product of two operators, we rewrite eq. (2.4)
as:
h
i
′
1
2
|Ψ i = R̂~u (α) ⊗ R̂~u (α) |φ(1)i ⊗ |φ(2)i
h
i
= R̂~u1 (α) ⊗ R̂~u2 (α) |Ψi .
~
~
~
R~u1 (α) ⊗ R~u2 (α) = e−i/~ αL1 ·~u e−i/~ αL2 ·~u = e−i/~ αL·~u ,
where
~ =L
~1 + L
~ 2,
L
47
2.4. ROTATION OF OBSERVABLES
h
i
~ 1, L
~ 2 = 0.
L
Reminder : let us consider E1 and E2 , two Hilbert spaces.
e
e
A(1)
A(2)
e = Â(1) ⊗ 1(2) ← extension of the operator A acting on E1 in E
A(1)
e
B(2)
= 1(1) ⊗ B̂(2)
Then, the tensor product of Â(1) and B̂(2) is:
e B(2).
e
Â(1) ⊗ B̂(2) = A(1)
Usually, in quantum mechanics the symbol ⊗ is omitted:
|φ(1)i |χ(2)i
means
|φ(1)i ⊗ |χ(2)i ,
Â(1)B̂(2) means Â(1) ⊗ B̂(2).
2.4
Rotation of observables
Up to now we considered rotation of states. We would also like to know how do observables rotate.
Consider an operator  with a discrete spectrum:
 |an i = an |an i .
We apply a rotation both to the physical system and to the measuring device so that
their relative position is unchanged. We rotate the state |an i:
|a′n i = R̂ |an i .
Then,
which means that:
 |an i = an |an i
→
→
→
Â′ |a′n i = an |a′n i ,
Â′ R̂ |an i = an R̂ |an i ,
R̂† Â′ R̂ |an i = an |an i
R̂† Â′ R̂ = Â
→
Â′ = R̂ÂR̂† .
In the special case of an infinitesimal rotation R~u (dα),
i
i ~
′
~
 =
1 − dαJ · ~u  1 + dαJ · ~u
~
~
h
i
i
= Â − dα J~ · ~u, Â + O(dα2).
~
We can classify the operators as:
Behavior under a
rotation
(2.5)
48
CHAPTER 2. THEORY OF ANGULAR MOMENTUM
Scalar observables
 is scalar if  = Â′
∀R̂. According to eq. (2.5), this implies that
h
i
Â, J~ = 0.
Examples of scalar observables: J~ 2 , or for a spinless particle ~r 2 , p~ 2 , ~r · p~.
Vector observables
A vector observable V̂ transforms as a vector:
h
i
~
V = V̂x , V̂y , V̂z .
Let us consider the component Vx . Vx is unchanged by a rotation about the Ox axis:
[Jx , Vx ] = 0.
If we perform a rotation Rêy (dα) about the Oy axis, the transform of Vx is (Vx )′ given by:
i
(Vx )′ = Vx − dα [Jy , Vx ] .
~
(2.6)
The vector êx transforms as:
ê′x = êx + dα êy × êx
= êx − dα êz
and, since (Vx )′ = V~ · ê′x ,
(Vx )′ = V~ · êx − dα V~ · êz
Vx − dα Vz .
=
(2.7)
Comparing eq. (2.6) and eq. (2.7), we see that
[Vx , Jy ] = i~Vz .
This can be generalized for a given rotation: an arbitrary infinitesimal rotation transforms
V~ · ~u into V~ · ~u′ , where ~u′ is the transform of ~u with respect to the rotation under
consideration.
r̂, p̂, L̂, Jˆ are vector operators.
2.5
Rotation invariance
The importance of rotations in physics is related to the fact that physical laws are rotationally invariant.
49
2.5. ROTATION INVARIANCE
2.5.1
Invariance of physical laws
Consider a physical system (S) (classical or quantum mechanical), which we subject to a
rotation R. If we simultaneously rotate all other systems or devices which can influence it,
the physical properties and behavior of (S) are not modified. This means that the physical
laws governing the system have remained the same: they are rotationally invariant.
There exist transformations, like the similarity transformation, with respect to which the
physical laws are not invariant. F.i., if we consider the hydrogen atom and multiply the
distance (proton-electron) by λ 6= 1, we obtain a system whose properties don’t remain
invariant.
Implications of rotation invariance:
What does it mean that the physical properties and behavior of the system are unchanged
by a rotation performed at time t0 ?
• Â′ (rotated from Â) has the same spectrum as Â. What implies that the probabilities
do not change and therefore R̂ is linear and unitary. All transformations which
leave the physical laws invariant are described by unitary operators, except for time
reversal, which is associated to an anti-unitary operator.
• the time evolution of the system is not affected:
|Ψ′ (t0 )i = R̂ |Ψ(t0 )i ,
|Ψ′ (t)i = R̂ |Ψ(t)i .
If |Ψ(t)i is a solution of the Schrödinger equation, R |Ψ(t)i is also a solution of this
equation, which implies that Ĥ is a scalar observable.
Let us consider an isolated system described by Ψ at t0 . A rotation R is performed on
the system:
|Ψ′ (t0 )i = R̂ |Ψ(t0 )i .
Let us evolve the rotated system in time:
|Ψ′ (t0 + dt)i = |Ψ′ (t0 )i +
dt
Ĥ |Ψ′ (t0 )i .
i~
If we had not performed the rotation, the state at t0 + dt would have been Ψ(t0 + dt):
|Ψ′ (t0 + dt)i = R̂ |Ψ(t0 + dt)i = R̂ |Ψ(t0 )i +
→
dt
R̂Ĥ |Ψ(t0 )i
i~
R̂Ĥ |Ψ(t0 )i = Ĥ |Ψ′ (t0 )i = Ĥ R̂ |Ψ(t0 )i
→
R̂Ĥ |Ψ(t0 )i = Ĥ R̂ |Ψ(t0 )i ,
from which follows that Ĥ commutes with all rotation operators. For this to be so, it
is a necessary and sufficient condition that Ĥ commutes with the infinitesimal rotation
generators:
~ = 0.
[Ĥ, J]
50
CHAPTER 2. THEORY OF ANGULAR MOMENTUM
The total angular momentum of an isolated system is a constant of motion.
We can choose for the state space a standard basis:
n
o
{|k, j, mi}
H, Jˆ2 , Jz
H |k, j, mi = E |k, j, mi ,
Jˆ2 |k, j, mi = j(j + 1)~2 |k, j, mi ,
Jz |k, j, mi = m~ |k, j, mi .
Exercise: Rotation of a diatomic molecule
2.6
Rotation operators for a spin 1/2 particle
~ and
Let us consider a non-relativistic spin-1/2 particle with orbital angular momentum L
~ The total angular momentum of the particle is
spin angular momentum S.
~ + S,
~
J~ = L
which should read
~ ⊗ 1 + 1 ⊗ S,
~
J~ = L
where 1 in the first term is the identity operator in the spin space, and 1 in the second
term is the identity operator in the infinite-dimensional ket space spanned by the position
eigenkets.
Decomposition of rotation operators into tensor products
In the state space of the particle, the rotation operator R̂~u (α) is associated with the
geometrical rotation R~u (α):
~
R̂~u (α) = e−i/~ αJ·~u .
~ acts only in Er , and S
~ acts only in Es , we can write R̂~u (α) as a tensor product:
Since L
(r)
(s)
R̂~u (α) = R̂~u (α) ⊗ R̂~u (α),
where
(r)
~
(s)
~
R~u (α) = e−i/~ αL·~u ,
R~u (α) = e−i/~ αS·~u .
Also,
|Ψi = |φi ⊗ |χi ,
h
i h
i
(r)
(s)
|Ψ′ i = R~u (α) |Ψi = R~u (α) |φi ⊗ R~u (α) |χi .
2.6. ROTATION OPERATORS FOR A SPIN 1/2 PARTICLE
51
Now, we calculate explicitly the rotation operators in Es .
~ = ~ ~σ,
S
2
~
R̂~u (α) = e−i/~ αS·~u = e−i 2 ~σ·~u .
α
We use the definition of the exponent of an operator:
(s)
R~u = 1 −
iα
1 α 2
1 α n
−i
−i
~σ · ~u +
(~σ · ~u)2 + . . . +
(~σ · ~u)n + . . . ,
2
2!
2
n!
2
(~σ · ~u)2 = ~u 2 = 1,
1
if n is even,
n
(~σ · ~u) =
~σ · ~u if n is odd.
Grouping together the even and the odd terms:
(−1)p
1 α 2
(s)
+ ...+
R~u (α) = 1 −
2! 2
(2p)!
α
1 α 3
− i~σ · ~u
−
+ ...+
2 3! 2
which gives:
(1/2)
Now, R~u
α 2p
+ ...
2
(−1)p α 2p+1
+ ... ,
(2p + 1)! 2
α
α
(s)
R~u (α) = cos 1 − i~σ · ~u sin .
2
2
(α) in the basis {|+i , |−i} is explicitly given as:
(1/2)
R~u (α)
=
cos α2 − iuz sin α2 (−iux − uy ) sin α2
(−iux + uy ) sin α2 cos α2 + iuz sin α2
.
(2.8)
Note that (due to the half-integer nature of the particles)
(1/2)
R~u
(2π) = −1
and
(1/2)
R~u
(0) = 1.
The fact that the spin state changes sign during a rotation through an angle of 2π is not
a problem since two state vectors differing only by a global phase factor have the same
physical properties. The way the observable  transforms
(1/2)
Â′ = R~u
the spectrum of  remains invariant.
†
(1/2)
(2π)Â R~u (2π) = Â
52
CHAPTER 2. THEORY OF ANGULAR MOMENTUM
2.6.1
Unitary unimodular group SU (2)
In the S = 1/2 subspace we derived the representation of the rotation operator which
acts on the two-component spinor χ (in the spin space) as a 2 × 2 matrix which has the
following properties. It is a unitary unimodular matrix
a
b
,
(2.9)
U(a, b) =
−b∗ a∗
where a and b are the complex numbers satisfying the unimodular condition,
|a|2 + |b|2 = 1,
and it is unitary:
†
U(a, b) U(a, b) =
a∗ −b
b∗ a
a
b
∗
−b a∗
= 1 (the number of independent real parameters is 3)
Comparing eq. (2.9) with eq. (2.8):
α
α
Re (a) = cos
, Im (a) = −uz sin
,
2
2
α
α
Re (b) = −uy sin
, Im (b) = −ux sin
.
2
2
a and b are called Cayley-Klein parameters. Historically, the connection between a unitary unimodular matrix and a rotation was known before the birth of quantum mechanics,
and in fact the Cayley-Klein parameters were used to characterize complicated motion of
gyroscopes.
The set of unitary unimodular matrices is a group which is known as
SU(2): S stands for special,
U stands for unitary,
2 stands for two dimensions.
In contrast, the group defined by multiplication operators with general 2 × 2 unitary matrices (not necessarily constrained to be unimodular) is known as U(2).
SU(2) is a subgroup of U(2)
Rotation of two-component spinors
E = Er ⊗ Es
Ψχ (~r) = h~r, χ |Ψi
|Ψ′ i = R̂ |Ψi
R̂ = R̂(r) ⊗ R̂(s)
Ψ′χ (~r) = h~r, χ |Ψ′ i = h~r, χ| R̂ |Ψi
53
2.7. ADDITION OF ANGULAR MOMENTA
Ψ′χ (~r)
=
XZ
χ′
d3 r ′ h~r, χ| R̂ |~r ′ , χ′ i h~r ′ , χ′ |Ψi
{|~r, χi} basis are tensor products:
h~r, χ| R̂ |~r ′ , χ′ i = h~r| R̂(r) |~r ′ i hχ| R̂(s) |χ′ i
h~r| R̂(r) |~r ′ i = R−1~r |~r ′ i = δ ~r ′ − (R−1~r)
(1/2)
hχ| R̂(s) |χ′ i = Rχχ′
X (1/2)
Ψ′χ (~r) =
Rχχ′ Ψχ′ (R−1~r)
χ′
Ψ′+ (~r)
Ψ′− (~r)
(1/2)
=
(1/2)
R++ R+−
(1/2)
(1/2)
R−+ R−−
!
Ψ+ (R−1~r)
Ψ− (R−1~r)
Exercise: Consider a neutron beam, which is perpendicular incident on a block of ferromagnetic material. What happens with its polarization?
2.7
Addition of angular momenta
In classical mechanics, if we have a system of N classical particles, the total angular
~ of this system with respect to a fixed point O is:
momentum L
~=
L
with
N
X
i=1
~i,
L
~ i = ~ri × p~i .
L
~
If the system of N interacting particles is isolated, then dL/dt,
the moment of the external
~ is a constant of motion,
force with respect to O, is zero, and the total angular momentum L
~
but not those of the particles individually, Li .
In quantum mechanics, how do we deal with many particles?
1) Let us consider two non-interacting particles in a central potential V (r):
H0 = H1 + H2 ,
with
H1 = −
~2
∆1 + V (r1 ),
2m1
~2
∆2 + V (r2 ).
H2 = −
2m2
From QM I we know that
h
i
~ i , H1 = 0 and
L
h
i
~ i , H2 = 0,
L
i = 1, 2
54
CHAPTER 2. THEORY OF ANGULAR MOMENTUM
~ 1 and L
~ 2 are constants of motion.
and L
2) We now assume that the two particles interact with a potential energy v(|~r1 − ~r2 |):
H = H1 + H2 + v(|~r1 − ~r2 |).
Then,
h
i h
i
~
~
L1 , H = L1 , v(|~r1 − ~r2 |) ,
~ 1 is not a constant of motion.
which is in general non-zero, hence, L
~ as
However, if we define the total angular momentum operator L
~ =L
~1 + L
~ 2,
L
its three components are constants of motion, f.i.,
[Lz , H] = [L1z + L2z , H]
~
∂v
∂v
∂v
∂v
=
x1
;
− y1
+ x2
− y2
i
∂y1
∂x1
∂y2
∂x2
since v depends only on
|~r1 − ~r2 | =
→
while
p
(x1 − x2 )2 + (y1 − y2 )2 + (z1 − z2 )2 ,
∂v
∂|~r1 − ~r2 |
x1 − x2
= v′
= v′
,
∂x1
∂x1
|~r1 − ~r2 |
x2 − x1
∂v
= v′
,
∂x2
|~r1 − ~r2 |
and therefore
[Lz , H] = 0.
~ L]
~ = 0, S
~ is also a constant of motion. BUT, relativistic
For particles with spin, since [S,
corrections introduce a spin-orbit coupling term of the form:
~ · S,
~
Hso = ξ(r)L
(we will explain this when we do relativistic QM)
~ and S
~ do not longer commute with the total Hamiltonian. For instance,
L
[Lz , Hso] = ξ(r)[Lz , Lx Sx + Ly Sy + Lz Sz ]
= ξ(r)(i~Ly Sx − i~Lx Sy )
or
[Sz , Hso] = ξ(r)[Sz , Lx Sx + Ly Sy + Lz Sz ]
= ξ(r)(i~Lx Sy − i~Ly Sx ).
55
2.7. ADDITION OF ANGULAR MOMENTA
If we set
~ + S,
~
J~ = L
the three components of J~ are constants of motion:
[Jz , Hso] = [Lz + Sz , Hso ] = 0.
We conclude that for each particle,
~1 + S
~1 ,
J~1 = L
~2 + S
~2 .
J~2 = L
and the total angular momentum,
J~ = J~1 + J~2 ,
n
o
commutes with the Hamiltonian of the system. We choose therefore H, J~ 2 , Jz as a
new basis. Since H commutes with J~ 2 and Jz , to work with this basis is easier than
to work with the basis {J~1 2 , J1z , J~2 2 , J2z } because neither J1z nor J2z commute with H.
Therefore, it is important to learn n
to add properly
angular momenta and perform a basis
o
2
2
change from {J~1 , J1z , J~2 , J2z } to J~ 2 , Jz , with J~ = J~1 + J~2 and Jz = J1z + J2z .
2.7.1
Addition of two spins 1/2
In order to introduce the addition rules for angular momentum, we first consider two
spin-1/2 particles (f.i., two electrons) with the orbital degree of freedom suppressed. The
total spin is written as:
~=S
~1 + S
~2 ,
S
or more rigorously:
We have that
~=S
~1 ⊗ 1 + 1 ⊗ S
~2 .
S
[S1i , S2j ] = 0,
while
[S1i , S1j ] = i~ǫijk S1k .
~2 .
and the same holds for S
Therefore,
[Sx , Sy ] = i~Sz ,
[Si , Sj ] = i~ǫijk Sk
~
for the total spin operator, S.
The eigenvalues of the various spin operators are denoted as follows:
~ 2 = (S
~1 + S
~ 2 )2
S
Sz = S1z + S2z
S1z
S2z
:
:
:
:
s(s + 1)~2
m~
m1 ~
m2 ~
56
CHAPTER 2. THEORY OF ANGULAR MOMENTUM
We can expand the ket corresponding to an arbitrary spin state of two electrons in terms
of:
~ 2 and Sz
1. eigenkets of S
S~1 2 and S~2 2 → {s, m} representation (singlet-triplet representation):
|s = 1, m = ±1, 0i
|s = 0, m = 0i
m = −s, . . . , s
~1z and S2z
2. eigenkets of S
2
2
S~1 and S~2
→ {m1 , m2 } representation:
|++i , |+−i ,
1 1
1 1
2 2 , 2 − 2 ,
|−+i , |−−i
11
1 1
−
2 2 , − 2 − 2
In each set, there are four base kets, and the relationship between set (1) and set (2) is:
|s = 1, m = 1i = |++i
since m = 1 implies that both spins have to be in m1 = 1/2 and m2 = 1/2 states.
Now applying the ladder operators and using:
p
S− |s, mi = (s + m)(s − m + 1)~ |s, m − 1i
and
S+ |s, mi =
p
(s − m)(s + m + 1)~ |s, m + 1i
we can obtain the next relations in the following way:
S− = S1− + S2− = (S1x − iS1y ) + (S2x − iS2y ),
S− |s = 1, m = 1i = (S1− + S2− ) |m1 = 1/2, m2 = 1/2i ,
p
s
1 1
1
1
− + 1 m1 = − , m2 =
(1 + 1)(1 − 1 + 1) |s = 1, m = 0i =
2 2
2
2
s
1 1
1 1
1
1
+
+
− + 1 m1 = , m2 = −
2 2
2 2
2
2
1 1
+
2 2
and
|s = 1, m = 0i =
1
√
2
m1 = 1 , m2 = − 1 + m1 = − 1 , m2 = 1
.
2
2
2
2
2.8. FORMAL THEORY OF ANGULAR-MOMENTUM ADDITION
57
Applying again S− we obtain:
1
1
|s = 1, m = −1i = m1 = − , m2 = −
.
2
2
|s = 0, m = 0i can be obtained by requiring it to be orthogonal to the other three kets:
1 1
1 1
11
1 1
+ a2 − −
+ a3 −
+ a4 −
,
|s = 0, m = 0i = a1 22
2 2
2 2
22
hs = 1, m = 1 |s = 0, m = 0i = 0
hs = 1, m = −1 |s = 0, m = 0i = 0
hs = 1, m = 0 |s = 0, m = 0i = 0
→
→
→
a1 = 0,
a2 = 0,
1
a3 = −a4 = √ ,
2
1
1 1
1
1
→ |s = 0, m = 0i = √ m1 = , m2 = −
− m1 = − , m2 =
.
2
2
2
2
2
The coefficients that appear on the right side are the Clebsch-Gordan coefficients. The
Clebsh-Gordan coefficients are the elements of the transformation matrix that connects
the {m1 , m2 } basis to the {s, m} basis. There is another way to derive these coefficients.
~ 2,
Suppose we write the 4 × 4 matrix corresponding to operator S
~1 · S
~2
~ 2 = S~1 2 + S~2 2 + 2S
S
= S~1 2 + S~2 2 + 2S1z S2z + S1+ S2− + S1− S2+ ,
using the {m1 , m2 } basis. This matrix is not diagonal
the ladder operators
because
connect different states, like, f.i., S1+ connecting − 12 12 with 21 21 . The unitary matrix
that diagonalizes this matrix transforms the |m1 , m2 i base kets into the |s, mi base kets.
The elements of this unitary matrix are the Clebsch-Gordan coefficients.
Exercise: Heisenberg model for two spin=1/2 particles H = S~1 · S~2
2.8
Formal theory of angular-momentum addition
We generalize the previous calculation to two angular-momentum operators J~1 and J~2 in
different subspaces. The components of J~1 (J~2 ) satisfy the commutation relations:
[J1i , J1j ] = i~ǫijk J1k
and
[J2i , J2j ] = i~ǫijk J2k ,
[J1k , J2l ] = 0
58
CHAPTER 2. THEORY OF ANGULAR MOMENTUM
between any pair of operators from different subspaces.
The infinitesimal rotation operator that affects both subspaces 1 and 2 is:
!
!
iJ~1 · ~u dα
iJ~2 · ~u dα
i(J1 ⊗ 1 + 1 ⊗ J2 ) · ~u dα
1−
⊗ 1−
=1−
.
~
~
~
The total angular momentum is:
J~ = J~1 ⊗ 1 + 1 ⊗ J~2 ,
which is more commonly written as:
J~ = J~1 + J~2 .
For a finite-angle rotation, we have:
R1~u ⊗ R2~u = exp
−iJ~1 · ~u α
~
!
⊗ exp
−iJ~2 · ~u α
~
!
.
The total J~ satisfies:
[Ji , Jj ] = i~ǫijk Jk .
J~ is the generator of the rotation for the entire system. For the choice of a basis set, we
have two options:
• Option 1: simultaneous eigenstates of J~1 2 , J~2 2 , J1z , and J2z denoted by
|j1 j2 ; m1 m2 i ;
• Option 2: simultaneous eigenstates of J~ 2 , J~1 2 , J~2 2 , and Jz denoted by
|j1 j2 ; jmi .
To see why it is so, we note that
[J~ 2 , J~1 2 ] = [J~ 2 , J~2 2 ] = 0,
and J~ 2 can be written as:
J~ 2 = J~1 2 + J~2 2 + 2J1z J2z + J1+ J2− + J1− J2+ .
Acting on |j1 j2 ; jmi with J~ 2 , J~1 2 , J~2 2 , and Jz , one then gets:
J~1 2 |j1 j2 ; jmi = j1 (j1 + 1)~2 |j1 j2 ; jmi ,
J~2 2 |j1 j2 ; jmi = j2 (j2 + 1)~2 |j1 j2 ; jmi ,
J~ 2 |j1 j2 ; jmi = j(j + 1)~2 |j1 j2 ; jmi ,
Jz |j1 j2 ; jmi = m~ |j1 j2 ; jmi .
Usually the states |j1 j2 ; jmi are written as |jmi.
2.8. FORMAL THEORY OF ANGULAR-MOMENTUM ADDITION
Note that
59
[J~ 2 , Jz ] = 0,
but
[J~ 2 , J1z ] 6= 0
and
[J~ 2 , J2z ] 6= 0,
which means that we cannot add J~ 2 to the set of operators in Option 1, or add J1z and
J2z to the set of operators in Option 2. Therefore, option 1 and option 2 are two maximal
sets of mutually compatible observables.
2.8.1
Clebsch-Gordan coefficients
Alfred Clebsch (1833-1872) and Paul Gordan (1837-1912) were two german mathematicians who worked with invariant theory, which is the theory that deals with the description of polynomial functions that do not change under transformations from a given linear
group.
We define the unitary transformation that connects bases |j1 j2 ; jmi and |j1 j2 ; m1 m2 i:
XX
|j1 j2 ; m1 m2 i hj1 j2 ; m1 m2 |j1 j2 ; jmi ,
(2.10)
|j1 j2 ; jmi =
m2
m1
where we use:
XX
m1
m2
|j1 j2 ; m2 m2 i hj1 j2 ; m1 m2 | = 1.
In eq. (2.10), the elements
hj1 j2 ; m1 m2 |j1 j2 ; jmi
are the Clebsch-Gordan coefficients.
Properties of the Clebsch-Gordan coefficients:
1. the coefficients vanish unless
m = m1 + m2 .
To prove this, we note that
Jz = J1z + J2z
and
(Jz − J1z − J2z ) |j1 j2 ; jmi = 0.
We multiply eq. (2.11) with hj1 j2 ; m1 m2 | on the left getting:
(m − m1 − m2 ) hj1 j2 ; m1 m2 |j1 j2 ; jmi = 0.
2. the coefficients vanish unless
|j1 − j2 | ≤ j ≤ j1 + j2 .
One can see this by visualizing J~ as the vectorial sum of J~1 and J~2 :
(2.11)
60
CHAPTER 2. THEORY OF ANGULAR MOMENTUM
Triangle selection rule: the coefficient
hj1 j2 ; m1 m2 |j1 j2 ; jmi can be different
from zero only if it is possible to form
a triangle with its three line segments
of lengths j1 , j2 , j
3. the Clebsch-Gordan (CG) coefficients form a unitary matrix and the matrix elements
are taken to be real by convention. As a consequence, the inverse coefficient of
hj1 j2 ; jm |j1 j2 ; m1 m2 i
is
hj1 j2 ; m1 m2 |j1 j2 ; jmi .
Since a real unitary matrix is orthogonal, we have the orthogonality condition:
XX
hj1 j2 ; m1 m2 |j1 j2 ; jmi hj1 j2 ; m′1 m′2 |j1 j2 ; jmi = δm1 m′1 δm2 m′2 ,
j
m
which follows from the orthogonality of |j1 j2 ; m1 m2 i and the reality of the CG
coefficients. Also,
XX
hj1 j2 ; m1 m2 |j1 j2 ; jmi hj1 j2 ; m1 m2 |j1 j2 ; j ′ m′ i = δjj ′ δmm′ .
m1
m2
When j ′ = j and m′ = m = m1 + m2 , we obtain:
XX
| hj1 j2 ; m1 m2 |j1 j2 ; jmi |2 = 1,
m1
m2
which is the normalization condition for |j1 j2 ; jmi.
4. possible notations for the CG coefficients:
• hj1 j2 ; m1 m2 |j1 j2 ; jmi;
• hj1 m1 j2 m2 |j1 j2 ; jmi;
• c(j1 j2 j; m1 m2 m);
• cj1 j2 (jm; m1 m2 );
• the CG coefficients can be also written in terms of Wigner’s 3-j symbols:
p
j1 j2
j
j1 −j2 +m
.
2j + 1
hj1 j2 ; m1 m2 |j1 j2 ; jmi = (−1)
m1 m2 −m
2.8. FORMAL THEORY OF ANGULAR-MOMENTUM ADDITION
2.8.2
61
Recursion relations for the Clebsch-Gordan coefficients
In this subsection, we will show that recursion relations together with normalization conditions almost uniquely determine all Clebsch-Gordan coefficients.
The coefficients with fixed j1 , j2 , and j and different m1 and m2 are related to each other
by recursion relations. In order to find some of them, let us investigate what results from
acting by the ladder operators J± on |j1 j2 ; jmi:
XX
|j1 j2 ; m1 m2 i hj1 j2 ; m1 m2 |j1 j2 ; jmi .
(2.12)
J± |j1 j2 ; jmi = (J1± + J2± )
m1
Using:
m2
p
J+ = |j, mi = (j − m)(j + m + 1)~ |j, m + 1i
p
J− |j, mi = (j + m)(j − m + 1)~ |j, m − 1i ,
and
we extend eq. (2.12) as:
p
(j ∓ m)(j ± m + 1) |j1 j2 ; j, m ± 1i =

X p

(j1 ∓ m′1 )(j1 ± m′1 + 1) |j1 j2 ; m′1 ± 1, m′2 i
m′1 m′2
+
X p
m′1 m′2

(j2 ∓ m′2 )(j2 ± m′2 + 1) |j1 j2 ; m′1 m′2 ± 1i × hj1 j2 ; m′1 m′2 |j1 j2 ; jmi .
We multiply the last equation by hj1 j2 ; m1 m2 | on the left and use orthonormality and
observe that non-vanishing contributions from the right-hand side are possible only with
m1 = m′1 ± 1 and
m2 = m′2
for the first term
and
m2 = m′2 ± 1
m1 = m′1
and
for the second term.
Accordingly, we obtain the following recursion:
p
(j ∓ m)(j ± m + 1) hj1 j2 ; m1 m2 |j1 j2 ; j, m ± 1i
p
(j1 ∓ m1 + 1)(j1 ± m1 ) hj1 j2 ; m1 ∓ 1, m2 |j1 j2 ; jmi
=
p
+
(j2 ∓ m2 + 1)(j2 ± m2 ) hj1 j2 ; m1 , m2 ∓ 1 |j1 j2 ; jmi .
Because the J± operators have shifted the m-values, the non-vanishing condition for the
CG coefficients is now:
m1 + m2 = m ± 1.
62
CHAPTER 2. THEORY OF ANGULAR MOMENTUM
LHS: left hand side;
RHS: right hand side.
We can graphically represent the importance of these recursion relations in a m1 m2 -plane.
The J+ relation tells us that the coefficient at (m1 , m2 ) is related to the coefficients at
(m1 − 1, m2 ) and (m1 , m2 − 1) as seen in the figure.
while the J− relations are as shown in the figure.
Using this graphical notation, one can obtain the various Clebsch-Gordan coefficients.
2.8.3
Clebsch-Gordan coefficients and rotation matrices
We have seen in the previous chapters that the angular momentum operators are the
generators of infinitesimal rotations.
Consider the rotation operators R̂(j1 ) and R̂(j2 ) . The product R̂(j1 ) ⊗ R̂(j2 ) is reducible in
the sense that after suitable choice of a basis, its matrix representation can take the form:
 (j +j )

R 1 2


R(j1 +j2 −1)
0


(j1 +j2 −2)


R

.


.
.


.
0
(j1 −j2 )
R
In the group theory notation, the product R̂(j1 ) ⊗ R̂(j2 ) is written as:
R(j1 ) ⊗ R(j2 ) = R(j1 +j2 ) ⊕ R(j1 +j2 −1) ⊕ · · · ⊕ R(j1 −j2 ) .
2.9.
63
BELL’S INEQUALITY
In terms of the matrix elements, it is written as:
(j )
(j )
R̂m11 m′ R̂m22 m′ =
1
2
XXX
j
m
m′
(j)
hj1 j2 ; m1 m2 |j1 j2 ; jmi × hj1 j2 ; m′1 m′2 |j1 j2 ; jm′ i R̂mm′ ,
where the j sum runs over from |j1 − j2 | to j1 + j2 .
2.9
Bell’s inequality
Since the simplest addition of angular momenta is that of two electron spins, let us discuss
spin correlations by considering a two-electron system in a spin-singlet state:
ST = 0.
In the |j1 j2 ; m1 m2 i basis,
|ST = 0i =
1
√
2
(|ẑ ↑, ẑ ↓i − |ẑ ↓, ẑ ↑i) ,
where the quantization direction is ẑ. F.i., in state |ẑ ↑, ẑ ↓i, electron 1 is in the spin-up
state and electron 2 is in the spin-down state.
Let us suppose that we make a measurement of the spin component of one of the electrons.
There is a 50-50 chance of getting either up or down. But if one of the components is
shown to be in the spin-up state, the other is necessarily in the spin-down state. When
the spin component of electron 1 is shown to be up, the measurement apparatus has
selected the first term |ẑ ↑, ẑ ↓i, and a subsequent measurement of the spin component
of electron 2 must ascertain that the state of the composite system is given by |ẑ+; ẑ−i.
This kind of correlation can persist even if the two particles are well separated and have
ceased to interact, provided that, as they fly apart, there is no change in their spin states.
This is, f.i., the case for a J = 0 system disintegrating spontaneously into two spin-1/2
particles with no relative angular momentum, since angular momentum must hold in the
disintegration process.
As an example, think of the decay of η mesons (mass 549 MeV/c2 ) into a muon pair:
η → µ+ + µ− ,
or a proton-proton scattering at low energies. The Pauli principle forces the proton to be
in a state
1
S0 (l = 0, s = 0),
and the spin state of the scattering protons must be correlated as above, even after they
get separated by a macroscopic distance.
Let us consider the situation shown in Fig. 2.2.
We will observe the following cases, for instance:
64
CHAPTER 2. THEORY OF ANGULAR MOMENTUM
Figure 2.2: The measurement apparatus
1. if A measures Sz of particle 1 and B measures Sx of particle 2, the correlation
between the two measurements is random;
2. if A measures Sx of particle 1 and B measures Sx of particle 2, there is a 100%
correlation between the two measurements;
3. if A makes no measurement then B’s measurements will show random results.
Orthodox quantum mechanical interpretation of this situation: a measurement is a filtration process. When Sz of particle 1 is measured to be positive, the component |ẑ ↑, ẑ ↓i is
selected. A subsequent measurement of the other’s particle Sz ascertains that the system
is in |ẑ ↑, ẑ ↓i.
Einstein’s locality principle and Bell’s inequality
Many physicists felt uncomfortable with this interpretation of spin-correlation measurements, like Albert Einstein.
Einstein’s locality principle: “The real factual situation of system S2 is independent of
what is done with system S1 , which is spatially separated from the former.”
Einstein-Podolsky-Rosen paradox : an alternative interpretation of QM argues that the
dynamic behavior at the microscopic level appears probabilistic because there are some
yet unknown parameters: the hidden variables.
The crucial point is whether alternative theories for QM make predictions different from
those of QM as a probabilistic theory
In 1964, J. S. Bell pointed out that the alternative theories based on Einstein’s locality
principle actually predict a testable inequality relation among the observables of spincorrelation experiments that disagrees with the predictions of the orthodox QM. Precision
experiments have conclusively established that Bell’s inequality is violated in such a way
that the orthodox quantum mechanical predictions are fulfilled within error bars.
For a nice discussion on this subject, please see: David Mermin, Physics Today (April
1985, 38) and Alain Aspect, Nature, Vol. 398, 189 (1999).