group theory application.pdf

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Applications

The most of important applications of group theory in physics arefound 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



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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

group of rotations in ordinary 3-dimensional space inducestransformations 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 directconsequence 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 0which is invariant under a

group symmetry transformation G.

In Dirac notation the energy eigenvalue equation is



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0|(0)>=(0)|0 > ……….. (a)

And the invariance of 0 is expressed by

0 .() = 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 0 we have,

00 > = 0|0 >

= 0

|0

>

= 0(()|0 >

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

eigenvalue 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 casewe 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 0 will correspond to an irreducible representation , say, of G,

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



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can be labeled as (0)

, where 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

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 (0)

are distinct.

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

„accidental‟ degeneracy (0)

(0)

with l <n, giving a degeneracy

= 2 . 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 symmetrygroup 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 Ψ is degenerate with Ψ (R is any

element of the symmetry group G). Unless Ψ = 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



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

two distinct sets of partners,

and , even if they form bases for the

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

themselves, and are not compelled by symmetry considerations to bedegenerate 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 1or 2. The eigenfunctions

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

operations 4and 2 . 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 1 and 2 . If we disregard possible accidentaldegeneracy, 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, and i, to each eigenfunction

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 to to characterize its behavior under rotation and inversion (by assigning it to the mth row of the l th

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 1 -

representation. The fact that they are pictured as having different

energies implies that the functions 1 1 and1

1 are linearly independent;

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

energy. Similarly, for the two levels labeled E, 1 2

are partners

which transform according to E and are thus necessarily degenerate;

1 2

are also partners, but the Ψ‟s and ‟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 anexact 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



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physics, where accurate calculations are only feasible when group

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



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

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 coulombforces 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. Theinteraction 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 lowestenergy. 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



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

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.



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

Cambridge University’ 1978.

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.

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