Contents

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2 CONTENTS

Introduction

Here we introduce a brief history of group theory, and discuss the relationbetween group theory and quantum mechanics.

Group theory studies the algebraic structures known as groups. TheEnglish word “algebra” is originated from Arabic work “al-jebra” meaning“reunion of breaking parts”. As a branch of pure mathematics algebrastudies the rules of operations and relations.

The birth of group theory in 1831 (E. Galois) was due to the effortsof many mathematicians, mainly in the following three areas:

1. Geometry.

A.F. Mobius in 1827, began to classify geometries using the fact thata particular geometry studies properties invariant under a particular trans-formation, although he was completely unaware of the group concept.

2. Number theory.

In 1761 L. Euler studied modular arithmetic. In particular he exam-ined the remainders of powers of a number modulo n. Although Euler’swork is not stated in group theoretic terms he does provide an exampleof the decomposition of an Abelian group into cosets of a subgroup. Healso proved a special case of the order of a subgroup being a divisor of theorder of the group. C. F. Gauss in 1801 was to take Euler’s work muchfurther and gave a considerable amount of work on modular arithmeticwhich amounts to a part of the theory of Abelian groups. He examinedorders of elements and proved (although not in this notation) that there isa subgroup for every number dividing the order of a cyclic group.

3. Algebraic equations.

Permutations were first studied by J.L. Lagrange in his 1770 paper onthe theory of algebraic equations. Although the beginnings of permutationgroup theory can be seen in this work, Lagrange never composes his per-mutations so in some sense never discusses groups at all. Cauchy played a

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major role in developing the theory of permutations. His first paper on thesubject was in 1815 but at this stage Cauchy is motivated by permutationsof roots of equations.

E. Galois was the first to really understand that the algebraic solu-tion of an equation was related to the structure of a group, le groupe despermutations. By 1832 Galois had discovered that special subgroups (nowcalled normal subgroups) are fundamental. He calls the decomposition ofa group into cosets of a subgroup a proper decomposition if the right andleft coset decompositions coincide.

About one century later, in 1930’s group theory was reformulated byWigner and Weyl in order to be applicable to quantum mechanics. In thisbook we follow the formalism of the two authors. In many cases grouptheory seems a more convenient method for solving problems in quantummechanics.

First we use an example to demonstrate some characteristics of grouptheory in solving quantum mechanics problem. The example is about oneparticle moving in an isotropic harmonic oscillator field.

A) The solution in Schrodinger theoryIn this method one solves the differential equation (the Schrodinger

equation), HΨ = EΨ, where Ψ = Ψ(x, y, z) is the wave function and H isthe Hamiltonian, which writes

H =1

2(p)2 + V (r) with V (r) =

1

2(r)2. (1)

The Hamiltonian can be written as

H =1

2(p21 + x2

1) +1

2(p22 + x2

2) +1

2(p23 + x2

3). (2)

By solving the above equation with boundary conditions (finite, conti-nuity,...etc), one obtains Ψ as follows,

Ψn1n2n3(x1, x2, x3) = Nn1n2n3Hn1(x1)Hn2(x2)Hn3(x3)e12(x2

1+x22+x2

3), (3)

where Hn(x) are the Hermite polynomials, i.e.

Hn(x) = (−1)nex2 ∂n

∂xn(e−x2

), (4)

CONTENTS 5

and the eigenvalue of energy is

E = n1 + n2 + n3 +3

2= N +

3

2. (5)

All other physical quantities can be calculated with the wave functionΨ(x1, x2, x3). In this method the solution process is almost the same as inclassical physics, except that for quantum mechanics there exist hypothesesthat lead to the Schrodinger equation and boundary conditions. We notethat for the application of Schrodinger theory, we must have a well definedpotential V (r).

B) The solution in Heisenberg theoryIn this solution one employs quantization and uses an algebraic method

(instead of calculus). We start from the Hamiltonian

H =1

2(p21 + p22 + p23) +

1

2(x2

1 + x22 + x2

3), (6)

where pi, xi are operators that obey the following commutation relation:

[xi, pj] = iδij, [xi, xj] = [pi, pj] = 0, (7)

which is the consequence of the particle-wave duality.Then we make the transformations:

aj =1√2(xj + ipj) and a†j =

1√2(xj − ipj). (8)

The operators a†j and aj obey the commutation relations:

[ai, a†j] = δij, [a†i , a

†j] = 0, [ai, aj] = 0. (9)

Since

a†1a1 =1

2(x1 − ip1)(x1 + ip1) =

1

2(x2

1 − ip1x1 + ix1p1 + p21)

=1

2(x2

1 + p21)−1

2i[p1, x1] =

1

2(x2

1 + p21)−1

2, (10)

etc, the Hamiltonian can be rewritten as

H = a†1a1 + a†2a2 + a†3a3 +3

2. (11)

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The meaning of a†1 can be revealed in the following. Suppose

H1|Ψn1 >= En1 |Ψn1 >

and a†1|Ψn1 > forms a new state which may have a different eigenvalue E′:

H1a1†|Ψn1 >= E ′a1

†|Ψn1 > .

One subtracts a1†H1|Ψn1 > from both sides of the equation and obtain

(H1a†1 − a†1H1)|Ψn1 >= (E ′ − En1)a

†1|Ψn1 > .

Since[H1, a

†1] = a†1,

one obtainsE ′ − En1 = 1 or E ′ = En1 + 1,

which means that the action of a†1 on |Ψn1 > increases the energy by a unit,i.e.

a†1|Ψn1 >−→ |Ψn1+1 > .

Therefore one can create all the states from the ground state (denotedas |0 >) by applying a†i repeatedly:

|Ψn1Ψn2Ψn3 >=3∏

i=1

(a†i )ni

√ni

|0 > . (12)

In this method commutators are employed, and a tedious calculus deriva-tion is avoided. The method looks more elegant than the previous one.However, one still needs to know in advance the analytical form of the po-tential, as is the case of the Schrodinger theory.

C) The solution in Group theoryStarting from the operators a†i and ai and the commutation relations

(eq.(9)) one can construct the generators Tij = a†iaj, which form a closedset under commutations such that

[Tij, Trs] = δjrTis − δisTrj. (13)

In terminology of group theory the operators Tij generate the U(3) group.The Casimir operator of the U(3) group is

C(U3) = T11 + T22 + T33, (14)

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which is the same as the Hamiltonian of harmonic oscillator (eq. (11)). Itis easy to show that C(U(3)) commutes with all the generators. i.e.

[C(U(3)), Trs] = 0, (15)

therefore, we claim that the harmonic oscillator system has the U(3) sym-metry. The physical quantities of the system can be expressed by thegenerators of this group. For example the angular momentum operatorL = x× p can be expressed as

L1 = x2p3 − x3p2 = i(a†3a2 − a†2a3) = i(T32 − T23).

The most important consequences of the U(3) symmetry is that there mayexist degeneracy in calculated spectrum as well as selection rules in tran-sitions.

In practice, one often solves the problem in a reverse way: startingfrom the experimentally observed degeneracy in energy and selection rulesin transition one can guess a symmetry (of a group G) for the system, andthen use the group-theoretical method to calculate physical quantities. Bycomparing the calculation with experiment data one can determine if theassumed symmetry is correct or not, or it needs a modification. A bigadvantage of the group theory method is that it does not require detailedknowledge on the Hamiltonian, which is not available in many branches ofmodern physics, especially in particle physics and nuclear physics. This isthe reason that the group theory is now applied to almost all the branchesof modern physics.

Due to the formulation of Wigner and Weyl, there exists the followingcorrespondence between quantum mechanics and group theory:

Group Theory Quantum MechanicsIrreps EigenstatesGenerators Physical QuantitiesThe Casimir Operator The Hamiltonian

Another feature of the group theory is that one group can be used todescribe several systems that are quite different in nature. The followingis one example:

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1. The SU(3) model for p-shell nucleiDue to observation of rotational bands, Elliott proposed the SU(3)

symmetry for rotational nuclei. In the p-shell nuclei there are three orbitalstates,

a†1|0 > a†0|0 > a†−1|0 >

l,m : (1, 1) (1, 0) (1,−1)

which are degenerate in energy. The SU(3) group is generated by thefollowing nine operators:

LM = (a†a)(1)M (M = 1, 0,−1)

QM = (a†a)(2)M (M = 2, 1, 0,−1,−2),

and the group has the subgroup chain

SU(3) ⊃ SO(3) ⊃ SO(2).

2. The SU(3) model for QuarksAccording to the degeneracy in the masses of hadrons, the SU(3) quark

model was proposed. One assume that the three types of quarks

u d s

a†u a†d a†s

are degenerate in energy. The SU(3) group is generated by the generatorsTij = a†iaj. The system is described by a subgroup chain

SU(3) ⊃ SU(2) ⊃ SO(2).

In view of the intimate relation between quantum mechanics and grouptheory we suggest that students with prior knowledge on quantum me-chanics take the group theory as an alternative mathematical formalism ofquantum mechanics.