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“Group Theory and its Applications in

Physics”

Submitted to:

Sir Imran Chaudhary

Prepared by:

Suleman Khalid

07020610-014

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Contents

Sr

No#

Contents Page

No#

1 Introduction 4

1.1 Definition 4

2 Development 5

3 Applications 7

3.1 Raising of Degeneracy 9

3.2 Classification of spectral terms 10

3.3 The solution of the Schrödinger

equation

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3.4 The classification of a state of

the systems of identical

particles

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3.5 Nuclear structure 14

3.6 Nuclear spectra in L-S coupling 14

4 References 16

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Abstract

The progress of Symmetry in Physics depends on the ability to separate

the analysis of a physical phenomenon into two parts. First, there are

the initial conditions that are arbitrary, complicated, and

Unpredictable. Then there are the laws of nature that summarize the

regularities that are independent of the initial conditions. The laws are

often difficult to discover, since they can be hidden by the irregular

initial conditions or by the influence of uncontrollable factors such as

gravity friction or thermal fluctuations.

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Group Theory

In mathematics and abstract algebra, group theory studies the

algebraic structures known as groups. More poetically,

“Group theory is the branch of mathematics that answers the

question, “What is symmetry?””

“A man who is tired of group theory is a man who is tired of life.”

Sidney Coleman

Symmetry principles play an important role with respect to

the laws of nature. They summarize the regularities of the laws that are

independent of the specific dynamics. Thus invariance principles

provide a structure and coherence to the laws of nature just as the laws

of nature provide a structure and coherence to the set of events.

Definition:

A group G may be defined as a set of objects or operations,

rotations, transformations, called the elements of G, that may be

combined, or “multiplied”, to form a well-defined product in G,

denoted by a *, that satisfies the following four conditions.

1) If a and b are any two elements of G, then the product a*b is also

an element of G, where b acts before a; or (a, b) a*b

associates (or maps) an element a*b of G with the pair (a, b) of

elements of G. this property is known as “G is closed under

multiplication of its own elements.”

2) This multiplication is associative: (a*b)*c=a*(b*c).

3) There is a unit element 1 in G such that 1*a=a*1=a for every

element a in G. The unit is unique: 1=1′*1=1′ .

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4) There is an inverse, or reciprocal, of each element a of G;

labeled1−1, such that a*1−1=1−1*a=1. The inverse is unique: if

1−1 and 1′−1 are both inverse of a, then

1′−1=1′−1*(a*1′−1)=( 1′−1*a)* 1−1=1−1 .[1]

Development:

One of the most important mathematical achievements of 20th

century was the collaborative effort, taking up more than 10,000

journal pages and mostly published between 1960 and 1980, that

culminated in a complete classification of finite simple groups.

The study of groups arose early in the nineteenth century in

connection with the solution of equations. Originally a group was a set

of permutations with the property that the combination of any two

permutations again belongs to the set. Subsequently this definition was

generalized to the concept of an abstract group, which was defined to

be a set, not necessarily of permutations, together with a method of

combining its elements that is subject to a few simple laws.

There are many physical systems whose underlying dynamics has

some symmetry. A good example is provided by the water molecule.

There is symmetry between the two hydrogen ions, which may be

interchanged without affecting the energy of the system. Again there is

translation symmetry: the interaction between any two ions situated at

two different positions depends only on their relative separation and

not on their absolute positions. That is, the potential energy is actually

a function of relative separation and same is the case for the kinetic

energy. Further more the system has a rotational invariance whereby

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its energy is independent of its absolute orientation. Thus underlying

Hamiltonian, the classical expression for the energy, which becomes an

operator in Quantum mechanics, is invariant under a set of

transformations of the coordinates, which includes reflections,

translations and rotations.

These different types of transformation all have the common

property that they form a group. The successive application of two

such transformations gives another one, which we call the product.

There is an identity transformation, which is simply to do nothing,

and each transformation has an inverse, which ‘undoes’ the

operation, i.e. the product of a transformation and its inverse is the

identity.

Any given particular planetary orbit is an ellipse. It is not even

circular, which worried the ancients greatly, but perhaps they should

have been more worried that even a circle does not respect the

spherical symmetry of the underlying Hamiltonian. The explanation

is that the symmetry has some how been broken by the initial

conditions of the motion, which picked out first of all a plane in

which the motion would take place, and then in that plane a

direction, say that of the semi-major axis. Paradoxically, the facts

that the motion remains in a plane and that the direction of the

major axis remains fixed are due to laws of angular momentum and

Rung-Lentz vector which are the consequence of the underlying

symmetries.[2]

Applications

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The most of important applications of group theory in physics are

found not in the classical mechanics but rather in quantum

mechanics. There the ground state of a system usually does exhibit

the full symmetry of the Hamiltonian, though a very important and

interesting exception to this occurs in the phenomenon of

spontaneous symmetry breaking, where, again because of some

uncontrollable perturbation of the initial conditions, one asymmetric

solution is picked out of an infinite set of possible ones. Thus the

fundamental interactions of the spins in a ferromagnetic are

rotationally symmetric, but when one is formed they align

themselves in some particular direction. But as an example of the

more usual scenario, consider the group state of the hydrogen atom;

the ground state wave function gives a spherically symmetric

probability distribution which indeed respects the spherical

symmetry of the 1/r potential.

As far as the excited states are concerned, the rotational

symmetry of the problem means that they can be classified by the

total angular momentum number l and the magnetic quantum

number m, which refers to the eigenvalue of its z component.

Moreover, the energy does not depend on m. this makes perfect

sense physically, since there is no preferred direction: the choice of z

axis was completely arbitrary. As far as mathematics is concerned, it

means that we have a degenerate space of eigenfunctions with m

ranging from +l to –l. which all have the same energy and can be

transformed into each other by rotations. We can take arbitrary

linear combinations of the 2l+1 eigenfunctions which are still

eigenfunctions with the same energy and total angular momentum.

They therefore form what is technically known as a vector space. The

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group of rotations in ordinary 3-dimensional space induces

transformations within this vector space, giving what is known as a

representation, which can be realized by matrices, in this case of

dimension (2l+1)*(2l+1).[3]

In classical mechanics the symmetry of a physical system leads to

conservation laws. Conservation of angular momentum is a direct

consequence of rotational symmetry, which means invariance under

spatial rotations. In the first third of 20th century, Wigner and others

realized that invariance was a key concept in understanding the new

quantum phenomena and in developing appropriate theories. Thus, in

quantum mechanics the concept of angular momentum and spin

momentum has become even more central.[4]

1) Raising of Degeneracy:

Degeneracy: We are typically concerned with the eigenvalues and

eigenvectors of a quantum Hamiltonian 𝐻0 which is invariant under a

group symmetry transformation G.

In Dirac notation the energy eigenvalue equation is

𝐻0|𝐸(0)>=𝐸(0)|𝐸 0 > ……….. (a)

And the invariance of 𝐻0 is expressed by

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𝐻0 . 𝑈(𝑔) = 0

Where U(g) is the unitary operator induced in the space of quantum

mechanical states by the physical transformation g.

Because of the invariance of 𝐻0 we have,

𝐻0 𝑢 𝑔 𝐸 0 > = 𝑈 𝑔 𝐻0|𝐸 0 >

= 𝑈 𝑔 𝐸 0 |𝐸 0 >

= 𝐸 0 (𝑈(𝑔)|𝐸 0 >

That is, U(g)| 𝐸 0 > is again an eigenstate of 𝐻0 with the same

eigenvalue 𝐸 0 transform among themselves under the action of the

group. They thus form a sub module in the complete space of

eigenvectors and provide the basis of a representation of G. it could be

that there is only one eigenstate with the given eigenvalue, in which case

we speak of a „non-degenerate‟ level. The representation is then just the

trivial representation. However, in many examples of physical interest

there is more than one such eigenstate, in which case we speak of the

level as being „generate‟. In the latter case the action of the group on the

space of degenerate states of the level induces an r-dimensional

representation, where r is the number of degenerate eigenvectors. In

general there is no reason to expect smaller invariant subspaces, which

means that the representation will be irreducible. Thus a given energy

level 𝐸 0 will correspond to an irreducible representation𝐷𝜇 , say, of G,

and the degeneracy r will be just the dimensionality 𝑛𝜇 of 𝐷𝜇 . The level

can be labeled as 𝐸𝛼𝜇(0)

, where 𝛼 comprises other labels not connected

with the group.[5]

An example which immediately springs to mind is the energy

spectrum of a particle moving in a central potential. The symmetry

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group is the 3-dimensional rotation group SO(3)., whose irreducible

representations, of dimension 2l+1, are labeled by the integer l

associated with the angular part of the wave equation. The principle

quantum number n, on the other hand, is associated with solutions of the

radial equation. For a general potential U(r) the levels 𝐸𝑛𝑙(0)

are distinct.

However, in the most familiar problem of all, U=-k/r, there occurs the

„accidental‟ degeneracy 𝐸𝑛𝑙(0)

𝐸𝑛(0)

with l<n, giving a degeneracy

𝑟 = 𝑛2 . This additional degeneracy, which means that each level

corresponds to a reducible representation of SO(3), arises from

invariance of the 1/r potential under the larger group SO(4).[8]

2) Classification of spectral terms:

If we are studying an atomic system, we must first find the symmetry

group of the Hamiltonian, i.e., the set of transformations which leave the

Hamiltonian invariant. The existence of a symmetry group for the

system raises the possibility of degeneracy. If Ψ is an eigenfunction

belonging to the energy ε, then 𝑂𝑅Ψ is degenerate with Ψ (R is any

element of the symmetry group G). Unless 𝑂𝑅Ψ = CΨ for all R, the

level is degenerate. The eigenfunctions belonging to a given energy ε

from the basis for representation of the group G. In most cases this

representation will be irreducible. Only in rare cases, for very special

choices of parameters, will we have “accidental” degeneracy, so that sets

of functions belonging to different irreducible representations coincide

in energy. It is clear that the partners who form the basis for one of the

irreducible representations of G must be degenerate, since they are

transformed into one another by operations of the symmetry group. But

two distinct sets of partners, 𝜓𝑖𝜇

and 𝜙𝑗𝜐 , even if they form bases for the

same irreducible representation of G(µ=𝜈), transformation only among

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themselves, and are not compelled by symmetry considerations to be

degenerate with one another.

So we may assume, in general, that the set of eigenfunctions

belonging to a given energy ε are a set of partners, and form the basis for

one of the irreducible representations of the symmetry group. This

already tells us a great deal about the degree of degeneracy to be

expected. For example, if we consider a system having the symmetry

group O, the energy level of the system can only be single, or doubly or

triply degenerate. The single levels will be of two types, depending on

whether they belong to the representations 𝐴1or𝐴2. The eigenfunctions

of these two types of simple levels differ in their behavior under the

operations 𝐶4 and 𝐶2 . The doubly degenerate levels will all be of the

same type, belonging to the two-dimensional representation E. finally,

there will be two different types of triply degenerate levels belonging to

the representations 𝐹1 and 𝐹2 . If we disregard possible accidental

degeneracy, these are only possible level types. Though the labels which

we use may appear strange, we are actually doing exactly what is done

in Quantum-mechanical treatments-we are assigning two quantum

numbers, 𝜇 and i, to each eigenfunction 𝜓𝑖𝜇

to describe its behavior

under the operations of the point-symmetry group. In the same way, as

we shall later see, when the symmetry group is the full rotation group,

we assign quantum numbers 𝑙 𝑎𝑛𝑑 𝑚 to 𝜓𝑚𝑙 to characterize its behavior

under rotation and inversion (by assigning it to the mth row of the lth

irreducible representation).

Thus the following level scheme might be typical for a system with

symmetry O:

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In this diagram two levels are drawn which belong to the 𝐴1 -

representation. The fact that they are pictured as having different

energies implies that the functions 𝜓1𝐴1 and𝜙1

𝐴1 are linearly independent;

if they were linearly related, they would necessarily have the same

energy. Similarly, for the two levels labeled E, 𝜓1𝐸𝑎𝑛𝑑 𝜓2

𝐸 are partners

which transform according to E and are thus necessarily degenerate;

𝜙1𝐸 𝑎𝑛𝑑 𝜙2

𝐸 are also partners, but the Ψ‟s and 𝜙 ‟s are linearly

independent of one another.[3]

3) The solution of the Schrödinger equation:

One of the most valuable application of group theory is to the

solution of the Schrödinger equation. Only for a small number of very

simple systems, such as the hydrogen atom, is it possible to obtain an

exact analytic solution. For all other systems it is necessary to resort to

numerical calculations, but the work involved can be shortened

considerably by the application of group representation theory. This is

particularly true in electric energy band calculations in solid state

physics, where accurate calculations are only feasible when group

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theoretical arguments are used to exploit the symmetry of the system

to the full.[5]

4) The classification of a state of the systems of identical

particles:

One of the main problems of atomic and nuclear physics is the

determination of identical (equivalent) particles. Since we cannot solve

the problem for a system of interacting particles, we use the methods

of perturbation theory. Each particle of the system is assumed to move

in some averaged potential field. We determine the eigenstates for this

average field and take, as basis functions for the full problem, products

of the single-particle field plus the interactions among the particles. If

the particles are identical, the interaction operator will be symmetric in

all the particles. Consequently its matrix elements between basis

functions will depend sensitively on the symmetry of these functions

under interchange of particles.[6]

5) Nuclear structure:

Perturbation procedures similar to those for the many-electron

problem can be applied to nuclei. The nuclear problem is complicated

by the fact that the system is built up from two kinds of particles,

neutrons and protons. (In addition, we have no definite knowledge of

the nuclear interaction. The comparison of calculated and observed

nuclear structures provides us with information concerning the nuclear

Hamiltonian.) The neutron and proton have (approximately) the same

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mass and spin (s=1/2), and transform into each other in beta-decay.

The neutron is neutral, while the proton has charge +e, so only the

protons will be subjected to Coulomb forces. However, the coulomb

forces are small compared to the specifically nuclear forces. In addition,

the available experimental evidence shows that the specifically nuclear

forces between two particles in the nucleus do not depend on whether

the particles are neutrons or protons-the nuclear forces are charge-

independent. It is therefore useful to regard neutron and proton as

state of a single fundamental entity which we call a nucleon.[7]

6) Nuclear spectra in L-S coupling:

If the nuclear forces do not depend strongly on the spins, we can,

as in the atomic problem, write the wave function as the product of an

orbital function and a function of the spin and charge variables. The

interaction Hamiltonian is symmetric in the space coordinates of the

nucleons, so the orbital wave functions should be combined to give a

total orbital function of definite symmetry. The energy of the state will

depend critically on this symmetry. Since the nuclear forces are

primarily attractive, the energy will be lowered if the symmetry of the

orbital wave function is increased. Thus we may expect that the state

whose orbital function has the highest symmetry will have the lowest

energy. Since the total wave function of the system of identical

nucleons is required by the Pauli principle to be completely anti

symmetric, we must construct charge-spin functions of definite

symmetry and obtain the total wave function by taking the product of

the orbital function with a charge-spin function having the conjugate

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symmetry. Since the energy of the state is determined only by the

orbital function, while the multiplicity depends on the charge-spin

function, each energy level will be a supermultiplet.[9]

Conclusion:

The concept of symmetry plays very important role in daily life

problems. Many physical systems are much complicated and still it is

impossible to completely solve the problems. On the bases of the group

theory the problem is reduced into groups by considering the symmetry

and then it becomes easy to solve. Same principle is used to solve the

Schrödinger equation and the most important applications are found in

Quantum mechanics. In short the symmetry concept has made very

easy to solve physical problems and its importance can also be seen in

other fields.

References:

1) Arfken & Weber, “Mathematical Methods For Physicists”,

Publisher, ‘ Elsevier Academic Press’ 2005.

2) Gene Dresselhaus,”Group Theory Applications to the Physics of

Condensed Matter”, Publisher ‘Springer’ 2007.

3) H. F. Jones, “Groups, Representations and Physics”, Publisher “ J

W Arrow smith”, 1998.

4) G. T. Hooft, “Lie Groups in Physics”, Publisher ‘Mc Graw Hill’,

2007.

5) John S, “ A course on Group Theory”, Publisher ‘Syndics of the

Cambridge University’ 1978.

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6) J. F. Cornwell, “Group Theory in Physics”, Publisher ‘Academic

Press London’ 1997.

7) M. Hamermesh, “Group Theory and its Applications to Physical

Problems”, Publisher ’Argonne National Laboratory’ 1959.

8) B. Baumslag Bruce C. “Group Theory, Schaum’s Outline Series”,

Publisher ‘Mc Graw Hill’ 1968.

9) Wu-Ki Tung, “Group Theory in Physics”, Publisher ‘World Sceince

Publishing Co. Pte. Ltd’ 1985.