Chapter 3 Spin and quantum mechanical rotation group

Chapter 3

Spin and quantum mechanicalrotation group

The Hilbert space of a spin 12 particle can be explored, for instance, through a

composition of Stern Gerlach experiments, ie. a composition of projectionoperators P (~e) which produce states with spin parallel to the polarisationvector ~e. From this exercise one concludes that the Hilbert space of a spin12 particle is a 2-dimensional vector space with complex coe�cients, C2

spanned by complex linear combinations of the vectors {|~e± >}. In particu-lar, if ~e and ~e 0 are 2 linearly independent polarisation vectors we have

|~e ✏ >=< +~e 0|~e ✏ > |~e 0+ > + < �~e0|~e ✏ > |~e 0� > (3.1)

The projection operator P (~e) can now be written as

P (~e) = |~e+ >< +~e| = |(�~e)� >< �(�~e)| (3.2)

Its matrix representation can be constructed noting that the most generalpure state (P 2 = P ) is given by

P =1

2

⇣

1 + ~✓ · ~�⌘

(3.3)

with |~✓| = 1 and

�i = {✓

0 11 0

◆

,

✓

0 �ii 0

◆

,

✓

1 00 1

◆

} (3.4)

are the Pauli matrices. It then follows that

P =1

2(1 + ~e · ~�) (3.5)

19

20CHAPTER 3. SPIN ANDQUANTUMMECHANICAL ROTATIONGROUP

3.1 Quantum mechanical rotation group

With the Hilbert space of a spin 12 particle constructed we now want to un-

derstand the action of the rotation group on this space. Let R 2 SO(3) be arotation in 3-dimensional Euclidean space. Then there is a 1� 1 correspon-dence

DR : P (~e) ! P (R~e) (3.6)

between pure states, P (~e) and elements of SO(3) which leaves expectationvalues invariant since

tr(P (~e)P (~f)) = tr(P (R~e)P (R~f)) (3.7)

and any linear operator acting on H = C2 can be expanded in terms ofprojectors P (~f), f 2 R3. Furthermore

DR1(DR2(P (~e))) = P (R1R2~e) = DR1R2(P (~e)) (3.8)

This provides a representation of SO(3) on {P (~f)}.We will now show that for each R 2 SO(3) there is a unitary transfor-

mation U(R) on H = C2 such that

DR(P (~e)) = U(R)P (~e)U⇤(R) (3.9)

Here ”*” stands for the hermitian conjugate. The rotation group SO(3) isLie group which means that any rotation R(~e,↵), can be written as

R(~e,↵) = ei↵~e·~K (3.10)

in terms of the 3 generators of infinitesimal rotations

K1 =

0

@

0 0 00 0 i0 �i 0

1

A , K2 =

0

@

0 0 �i0 0 0i 0 0

1

A , K1 =

0

@

0 i 0�i 0 00 0 0

1

A ,

(3.11)

defining the Lie algebra, so(3),

[Ki, Kj] = i✏ kij Kk (3.12)

3.1. QUANTUM MECHANICAL ROTATION GROUP 21

These relation, together with (3.11) furthermore imply that SO(3) has aninterpretation of a di↵erentiable (group) manifold.

In addition to the defining representation (3.11), this Lie algebrahas infinitely many irreducible representations and, in particular, a 2-dimensional representation, D 1

2in terms of the Pauli matrices (3.4) through

~S =1

2~� (3.13)

For instance, S3 measures the spin in the z-direction

~S3|~ez± >= ±~2|~ez± > (3.14)

where ~ has been included for dimensional reasons. Using

[Si, Sj] = i✏ kij Sk (3.15)

we then conclude that a rotation R(~e,↵), is implemented on C2 as

U(~e,↵) = e�i↵~e· ~S (3.16)

The group generated by these matrices is denoted by SU(2) since det(U) = 1(special) and U⇤ = U�1 ( unitary). In particular,

U(~e, 2⇡)|~ez+ >= e�i⇡�3 |~ez+ >= �|~ez+ > (3.17)

which shows that U(~e,↵) is a projective representation of R(~e,↵). Due tothe identity (exercise)

R(~e,↵) ~S = U(~e,↵)⇤ ~SU(~e,↵) (3.18)

there is a unique element of in SO(3) for any U 2 SU(2) (homeomorphically)with

R(U1U2) = R(U1)R(U2) (3.19)

This means that SO(3) is a representation of SU(2) but not vice-versa. Inmathematics this property is sometimes expressed as a short exact se-quence 1

1 ! Z2 ! SU(2) ! SO(3) ! 1 (3.20)

1Here the !’s indicates that the image of each map is in the kernel of the followingmap.


Geometric interpretation: In addition to the Lie-algebra relations,the Pauli matrices satisfy the anti-commuation relations

{�i, �j} = �i�j + �j�i = 2�ij1 (3.21)

so that any element U(~e,↵) can be written in terms of {1,~�} as

U(~e,↵) = e�i2↵~e·~� = cos(

↵

2)1� i(~e · ~�) sin(↵

2) =: a01+ ~a~� (3.22)

with a0 2 R, ~a = �i~↵, ~↵ 2 R3, so that a20 + |~↵2| = 1. Therefore thegroup generated by U(~e,↵) has the same topology as the 3-sphere, S3. Incontrast to the group manifold of SO(3), the quantum mechanical rotationgroup defines a simply connected group manifold2. This group is thereforethe universal covering group of SO(3).

3.2 Product representations

The Hilbert space of two distinguishable spin 12 particles can be explored,

for instance, with the help of two independent Stern-Gerlach experiments.The space is spanned by the product states

|~e 1✏1,~e 2✏2 >⌘ |~e 1✏1 > |~e 2✏1 >= |~e 1✏1 > ⌦|~e 2✏1 > (3.23)

with scalar product

< ✏1 ~f 1, ✏2 ~f 2|~e 1✏1,~e 2✏2 >=< ✏1 ~f 1|~e 1 ✏1 >< ✏2 ~f 2|~e 2 ✏2 > (3.24)

which implies statistical independence. In what follows we will assumethat ~e 1 = ~e 2 and denote the states simply by |±,± >. It is important tonote that not all pure states in H are a product states. For instance for thestate vector

1p2(|+,� > �|+,� >) 2 H (3.25)

there is statistical correlation or entanglement between the two spin 12

particles which was the origin of the EPR paradox (see chapter ??).

2The SO(3) group manifold is given by the 3-sphere with antipodal points identified.

3.3. OBSERVABLES 23

3.3 Observables

Observables of the combined system are given by linear combinations ofproduct observables

A = A1 ⌦ A2

A1 ⌦ A2|✏1 ✏2 > = A1|✏1 > ⌦A2|✏2 > (3.26)

for example,

S13 |±,+ >⌘ S1

3 ⌦ 1|± > |+ >= ±~2|±,+ > (3.27)

Since [S1i , S

2j ] = 0 the total spin ~S ⌘ ~S 1 + ~S 2 satisfies the Lie-algebra

relations (3.15). This is a product representation denoted by D 12⌦ D 1

2.

The representationD 12is irreducible since there are no invariant subspaces

under the action of all possible polynomials of Si. On the other hand, D 12⌦D 1

2

is reducible. It can be decomposed into a direct sum of irreducible singletand triplet representations

D 12⌦D 1

2= D0 �D1 (3.28)

with

D0 = { 1p2(|+,� > �|+,� >)} (3.29)

D1 = {|++ >,1p2(|+,� > �|+,� >) |�� >} (3.30)

More generally there is an irreducible representation Dj of SU(2) for allvalues of j = 0, 12 , 1,

32 , · · · with dimension dim(Dj) = 2j + 1 and

( ~S)2| >= j(j + 1)| >, 8 | >2 Dj (3.31)

In addition any representation of SU(2) can be decomposed into into a directsum of irreducible representations, Dj.


3.4 Clebsch Gordan Decomopsition

Let Dj1 and Dj2 be two irreducible representations of SU(2) with ji =0, 12 , 1,

32 , · · · then we have

Dj1 ⌦Dj2 = Dj1+j2 �Dj1+j2�1 � · · ·�D|j1�j2| (3.32)

with every irreducible representationsDj appearing exactly once. If {|j1,m1 >} and {|j2,m2 >} is the standard basis in Dj1 and Dj2 respectively, thenthe elements {|(j1, j2); j;m >} of the standard basis of Dj is related to{|j1,m1 >} and {|j2,m2 >} through

|(j1, j2); j,m >=X

m1,m2

< m2, j2| < m1, j1|(j1, j2); j,m > |j1,m1 > |j2,m2 >

(3.33)

with j = j1 + j2, · · · , |j1 � j2|. The Clebsch Gordan coe�cients

< m2, j2| < m1, j1|(j1, j2); j,m >⌘< m2, j2,m1, j1|j,m > (3.34)

are elements of a unitary matrix which can furthermore be chosen orthog-onal such that

|j1,m1 > |j2,m2 >=X

j,m

< m2, j2,m1, j1|j,m > |j,m > (3.35)

i.e.X

j,m

< m2, j2,m1, j1|j,m >< m, j|j1,m01, j2,m

02 > = �m1m0

1�m2m0

2

X

m1,m2

< m, j|j1,m1, j2,m2 >< m2, j2,m1, j1|j0,m0 > = �jj0�mm0

(3.36)

and

< m2, j2,m1, j1|j,m >= 0 if m 6= m1+m2 or j /2 {j1+j2, · · · , |j1�j2|}(3.37)

3.5. IRREDUCIBLE TENSOR OPERATORS 25

Example: For j1 � 12 and j2 =

12 we have

|j1 + 1

2,m > =

s

j1 +m+ 12

2j1 + 1|m� 1

2> |1

2> +

s

j1 �m+ 12

2j1 + 1|m+

1

2> |� 1

2>

|j1 � 1

2,m > = �

s

j1 �m+ 12

2j1 + 1|m� 1

2> |1

2> +

s

j1 +m+ 12

2j1 + 1|m+

1

2> |� 1

2>

(3.38)

3.5 Irreducible Tensor Operators

Let D be an irreducible representations of SU(2) on H. A collection of 2j1operators, Tjm, m = j, · · · ,�j on H will be normal components of anirreducible tensor operator Tj of type j if they transform under g 2SU(2) as

U(g)TjmU(g)⇤ =X

m0

U(g)(j)mm0Tjm0 (3.39)

where U(g)(j)mm0 are the matrix components, in the standard basis, of the spinj representation Dj of g. The transformation property (3.39) is equivalent to

[K3, Tjm] = mTjm (3.40)

[K±, Tjm] =p

j(j + 1)�m(m+ 1)Tjm±1 (3.41)

where the {Ki} generate the representation of (3.12) on H. An example fora scalar operator is the Hamiltonian H. The spin operators {S3, S± =S1 ± iS2} are the normal components of a vector operator of type 1.

Let H(j1)⌧1 , H(j2)

⌧2 be subspaces of H carrying the representations Dj1 andDj2 of SU(2) respectively with possibly further quantum numbers indicatedby ⌧ . Then the matrix elements of a tensor operator Tj in the standard basis|⌧, j,m > have the simple form

< ⌧2, j2,m2|Tjm|⌧1, j1,m1 >=< ⌧2, j2||Tj||⌧1, j1 >< m1, j1,m, j|j2,m2 >

(3.42)

where< m1, j1,m, j|j2,m2 > are the CGK-coe�cients and< ⌧2, j2||Tj||⌧1, j1 >is the reduced matrix element. This is theWigner-Eckhardt-Theorem.We will make use of it when considering the interaction of atoms with theradiation field.


Historical Notes

Stern had a PhD in Chemistry and became a pupil of Einstein in Prag wherehe learned about Bohr’s quantization rule. Stern and his colleague Max vonLaue made a vow: ”If this nonsense of Bohr should in the end prove to beright, we will quit physics!” Experimental progress was hampered by financialdi�culties. So Stern presented a series of public lectures with an entrancefee. He also received a cheque for some hundreds of dollars. Goldman, afounder of the investment firm Goldman Sachs and progenitor of WoolworthCo stores, had family roots in Frankfurt. Because of frequent breakdown ofthe apparatus the deposition of silver atoms of the plate was very thin. Whilelooking at the plate Stern was pu�ng a cheap cigar which contained a lot ofsulfur. This sulfur developed the silver into silver sulfide which is more easilyvisible. Pauli’s reaction to the result of the Stern-Gerlach experiment was:This convinced me once and for all that an ingenious classical mechanismwas ruled out to explain atomic phenomena...

Further Reading

A good text book for the quantum mechanical rotation group is Sakurai’sModern Quantum Mechanics, chapter 3.

For a concise mathematical treatment of the representation theory of therotation group and the Lorentz group see e.g. V. S. Varadaraja, ”Supersym-metry for mathematicians: an introduction” , Volume 11 of Courant lecturenotes, section 1.4 and 1.5, available online under http://books.google.com/

A detailed discussion of the Wigner-Eckhardt-Theorem can be found inH. Rollnik, ”Quantentheorie”, Bd. 2.

A readable book on some aspects of group theory is Wu-Ki-Tung, ”GroupTheory in Physics”.

Chapter 4

Quantum mechanical Lorentzgroup

Our analysis of the quantum-mechanical rotation group lead to the inter-pretation of the rotation group on R3 as a representation of the quantum-mechanical rotation group SU(2) through the important relation

A 2 SU(2) =) A~e · ~�A⇤ = (R(A)~e) · ~� (4.1)

with A⇤ = A�1. If we introduce a ”fourth” Paul matrix �0 =

✓

1 00 1

◆

we

have similarly in the relativistic four-vector notation1

Aqµ�µA⇤ = R(A)µ⌫q

⌫�µ (4.2)

with R(A)0⌫ = 0 since A�0A⇤ = �0. However, if we instead consider A 2SL(2, C), the group of all complex 2 ⇥ 2-matrices with unit determinantthe situation changes since now A⇤ 6= A�1. More precisely

A : Q ⌘ qµ�µ =

✓

q0 + q3 q1 � iq2

q1 + iq2 q0 � q3

◆

! AQA⇤ ⌘ AQ (4.3)

The map Q ! AQ is linear, satisfies AQ = AQ⇤ and

detQ = �qµqµ = detAQ = �AqµAqµ (4.4)

1We we assume the convention ⌘µ⌫ = diag(�1.1.1.1).

27

28 CHAPTER 4. QUANTUM MECHANICAL LORENTZ GROUP

Thus there exists a Lorentz transformation ⇤(A) 2 SO(3, 1) such that

Aqµ�µA⇤ = (⇤(A))µ⌫q

⌫�µ (4.5)

In other words, the map A ! ⇤(A) is a homomorphism from SL(2,C)onto the proper, orthochronous Lorentz transformations SO+(3, 1) withkernel {1,�1}. It the follows form the first isomorphism theorem ingroup theory that SL(2,C) is isomorphic to SO+(3, 1)/{1,�1}.

4.1 Spinor representations

We will see that the representation theory of SL(2,C) leads to the correctrelativistic quantum theory for spin 1

2 particles. For this we first consider the2-dimensional representation V with (↵, � = 1, 2)

A : V 3 u ! Au i.e. 0u↵ = A↵�u

� (4.6)

An element in the dual space, V then transforms with AT�1 ⌘ A �↵

A : V 3 v ! AT�1v i.e. 0v↵ = A �↵ v� (4.7)

Consequently the ”scalar product” v�u� = 0v� 0u� is an invariant in analogywith qµqµ. In Minkowski space-time the metric provides an isomorphismbetween V and its dual. In the spinor calculus this isomorphism is providedby the invariant tensor

✏↵� ⌘ (i�2)↵� =

✓

0 1�1 0

◆

(4.8)

or

1 = ✏12 = �✏21 = 1 = �✏12 = ✏21 (4.9)

and ✏↵↵ = 0. Under an SL(2,C)-transformation we have

0✏↵� = A↵�A

��✏

�� () 0✏ = A✏AT = ✏ (4.10)

The last identity is the result of an explicit calculation. As a consequencewe have

AT�1 = ✏�1A✏ (4.11)

4.2. PARITY 29

In analogy with Minkowski space-time we can think of v� and v� as the con-travariant and the covariant components of a spinor in V . Note, however thatv�u� = �v�u� which shows, in particular, that spinors cannot be representedin tems of commuting objects in a relativistic quantum theory.

In addition to the representation just discussed we there is a second in-equivalent representation V , the complex conjugate or anti-fundamentalrepresentation with (↵, � = 1, 2)

A : V 3 u ! Âu i.e. 0u↵ = Â↵�u� (4.12)

where ”¯” denotes complex conjugation. Similarly with A⇤�1 = ¯AT�1 ⌘ Â �↵

A : ˜V 3 u ! A⇤�1u i.e. 0u↵ = Â �↵ u� (4.13)

with invariant product u�u� and ✏↵� ⌘ (i�2)↵� e.t.c. Thus the complex

conjugate components of a spinor in V transforms like a spinor in V

4.2 Parity

Next we will investigate how parity-symmetry is represented on spinors. Forthis let us now return to the object Q in (4.3). From its transformationproperty (4.3) we infer that Q is an element of V ⌦ V , ie.

Q = Q↵� = qµ(�µ)↵� (4.14)

In other words Q is an element of the product representation denoted by(12 , 0)⌦(0, 12) = (12 ,

12), which, according to (4.14) must be the vector repre-

sentation. Note that the two representations (12 , 0) and (0, 12) are identicalfor the SU(2)-subgroup of SL(2,C). However they are inequivalent forSL(2,C) since there is no generator in SL(2,C) which takes (12 , 0) to (0, 12).

Under a parity transformation

P : q = (q0, ~q) ! P q = (q0,�~q) (4.15)

the bi-spinor Q transforms as

PQ = q0 � ~q · ~� = ✏�1Q✏ ⌘ qµ(P�µ) (4.16)


4.3 Weyl/Dirac equation

Using the anti-commuation relations (3.21) it is not hard to see that

PQQ = QPQ = �qµqµ (4.17)

Weyl considered a massless electron for which the four momentum pµ satisfiesp2 = 0. Then the two equations

p↵� v�(p) = pµ�↵�µ v�(p) = 0 (4.18)

p↵�u�(p) = pµ(P�µ)↵�u

�(p) = 0 (4.19)

define relativistic covariant Schrodinger equations in momentum space. Inposition space we have accordingly

i~@µ�↵�µ v�(x) = 0 and i~@µ(P�µ)↵�u�(x) = 0 (4.20)

Note that solutions of (4.20) are NOT of the form �(x)�↵.If the mass M is not vanishing we note that

p↵� v�(p) = Mc u↵(p) and p↵� u�(p) = Mc v↵(p) (4.21)

imply upon using (4.17) the Klein-Gordon equation in momentum space

(p2 +M2c2)v↵(p) = 0 (4.22)

and analogously for u�(p). The two equations (4.21) constiute the Diracequation for u↵ and v↵.

4.4 Coupling to the electro-magnetic field

We obtain gauge-invariant equation in the presence of an electro-magneticfield through the substitution2

i~@µ ! i~@µ � e

cAµ (4.23)

2In or conventions the relation between Aµ and the Coulomb– and vector potential isgiven by Aµ = (��,� ~A).

4.4. COUPLING TO THE ELECTRO-MAGNETIC FIELD 31

Equivalently with A↵� = A0 + ~A · ~� and PA↵� = A0 � ~A · ~� the covariantDirac equations

(i~@↵��e

cA↵�) v�(x) = Mc u↵(x) and (i~@↵��e

cPA↵�) u

�(x) = Mc v↵(x)

(4.24)

are gauge invariant since under a gauge transformation Aµ(x) ! Aµ(x) �@µ�(x) and

u↵(x) ! eie~c�(x)u↵(x) , v↵(x) ! e

ie~c�(x)v↵(x) (4.25)

the form of these equations is preserved.In practice it is often convenient to express the coupled system of the

2-component spinors u↵(x) and v↵(x) as a single equation for a 4-componentspinor, i.e. with

=

✓

uv

◆

, �µ =

✓

0 �µP�µ 0

◆

, A/ = Aµ�µ =

✓

0 APA 0

◆

, @/ = @µ�µ

(4.26)

we obtain the compact equation

(i~@/� e

cA/) = Mc (4.27)

The four Dirac matrices �µ satisfy the Cli↵ord algebra

{�µ, �⌫} = �2⌘µ⌫ (4.28)

Under Lorentz transformations 0xµ = ⇤µ⌫x

⌫ the 4-component spinor trans-forms reducibly

0 (0x) = S(A) (x), S(A) =

✓

A 00 (A⇤)�1

◆

(4.29)

Note that the representation (4.26) of the Cli↵ord algebra is unique only up

to linear transformations on

✓

uv

◆

. In particular we can choose any set of

Dirac matrices �µ with

{�µ, �⌫} = �2⌘µ⌫ (4.30)


A result due to Pauli shows that �µ = S�1�µS and thus

(i~@/� e

cA/) = Mc =) (i~@/� e

cA/) = Mc (4.31)

A useful choice for analysing the non-relativistic limit of the Dirac equationis the so-called standard basis used in problem sheet 7.

The quantization of the relativistic spin 1/2 particle proceeds in closeanalogy with the quantization of the electro-magnetic field explained in chap-ter 2 by expanding the general solution of the Dirac equation in terms of”plane waves” with fixed four momentum k, multiplied with (anti-commuting)operator valued coe�cients b(k) and b†(k).

Further Reading

A detailed discussion of the spinor representation of the Lorentz group canbe found in e.g.Y. Ohnuki, Unitary Representations of the Poincare Group

and Relativistic Wave Equations, chapter 2, available on line with Googlebooks. A description of the quantization of the Dirac field (4.26) can befound Sakurai, Advanced Quantum Mechanics, section 3.

Documents

Chapter 3 Spin and quantum mechanical rotation group