Prolog

Text Books: – W.K.Tung, "Group Theory in Physics", World Scientific (85)

– J.F.Cornwell, "Group Theory in Physics", Vol.I, AP (85)

Website: http://ckw.phys.ncku.edu.tw

Homework submission: class@ckw.phys.ncku.edu.tw

Grades:Exercises: 50%MidTerm: 20%Final: 30%

Why Group Theory?

• Physical problems ~ differential equations.

• Solutions to DE ~ linear vector / Hilbert space V.

• Symmetries in physical system → structures in V.

• → Group theory.

Applications

Finite / discrete / countable groups:• Molecules, crystals, defects, ions, surface absorbances,…• Spectral analysis: Electronic, vibrational, rotational, … • Obtain degeneracies & classifications of eigenstates effortlessly.• Ditto selection rules & branching ratios.• Obtain block diagonalized hamiltonians & symmetrized bases.

Lie (continuous) groups:• Rotation, Lorentz, Poincare groups:

Angular momentum, spin, ...• Classification of elementary particles:

Isotopic multiplets for hadrons / quarks.• Construction of gauge fields & unified theories.

• Electroweak: SU(2)U(1)• Strong: SU(3)• Unified: SU(5)

Properties of special functions.

Group Theory in Physics

1. Introduction2. Basic Group Theory3. Group Representations4. General Properties of Irreducible Vectors & Operators5. Representations of the Symmetric Groups6. 1- D Continuous Groups7. Rotations in 3-D Space: The Group SO(3) 8. The Group SU(2) & SO(3)

9. Euclidean Groups in 2- & 3-D Spaces10. The Lorentz & Poincare Groups, & Space: Time Symmetries11. Space Inversion Invariance12. Time Reversal Invariance13. Finite–Dimensional Representations of the Classical Groups

W.K.Tung, World Scientific (85)

Appendices

I. Notations & Symbols

II. Summary of Linear Vector Spaces

III. Group Algebra & the Reduction of Regular Representations

IV. Supplement to the Theory of the Symmetric Groups

V. Clebsch-Gordan Coefficients & Spherical Harmonics

VI. Rotational & Lorentz Spinors

VII. Unitary Representations of Proper Lorentz Group

VIII. Anti-Linear Operators

Supplementary Readings

• Similar to Tung but more physically oriented:– M.Hamermesh, "Group Theory & its Application to Physical

Problems", Addison-Wesley (62)

• Molecular theories: – F.A.Cotton, "Chemical Applications of Group Theory", 2nd ed.,

John Wiley (71)

• Teasers on atomic, molecular & solid-state applications:– M.Tinkham, "Group Theory & Quantum Mechanics", McGraw Hill

(64)

• Solid state theories:– T.Inui, et al, "Group Theory & its Applications in Physics",

Springer-Verlag (90)

1. Introduction

1. Particle on a 1-D Lattice

2. Representations of the Discrete Translation Operators

3. Physical Consequences of Translational Symmetry

4. The Representation Functions and Fourier Analysis

5. Symmetry Groups of Physics

Aim:

To demonstrate relations between

physical symmetries,

group theory,

& special functions ( group representation )

1.1. Particle on a 1-D Lattice

2

2

pH V x

m V x nb V x

b = Lattice constant

Translational symmetry (discrete): H invariant under x x n x x nb T

T n Effect on states:

{ |Ψ } & { |Ψ } physically equivalent

†A T n A T n

†T n T n

†A T n AT n

A

Symm ops are unitary

Physical operators are linear Symmetry operators are linear

†T n

†1T n T n

x n x x nb T x T n x x nb

†V x T n V x T n

V x nb

x V x x

V xx x

x V x a x

V x ax x

V x V x

x V x xV x

x x By def:

V x nb

x V x xV x nb

x x

x V x nb x

x x

1V n x T

Transl symm V x V x V x nb V x

Similarly:

p x p x nb i x nb

i x

p x

2

2

p xH x V x

m

H x †T n H x T n

H T n T n H , 0T n H

Eigenstates of T(n) is also eigenstates of H

x V x x

V xx x

T n V x V x T n

1.2. Representations of the Discrete Translation Operators

{ T(n) } or any set of symm ops satisfies ( dropped ) :

T n T m T n m Closure

0T E Existence of Identity

1T n T n

Existence of Inverse

T n T m T k T n T m T k Associativity

Td = { T(n) } forms a group

Furthermore: T n T m T m T n Td is Abelian ( commutative)

†T n T n E Td is unitary

More accurately: { T(n) } is the realization of Td = { T(n) } on the single particle Hilbert space .

, 0T n T m → simultanous eigenvectors | ξ T(n)

nT n t tn(ξ ) = eigenvalue of T(n) associated with | ξ

T n T m T n m

0T E

1T n T n

T n T m T k T n T m T k n m k n m kt t t t t t

0 1t

n m n mt t t

1n nt t

T n T m T m T n

†T n T n E

n m m nt t t t

* 1n nt t

Ansatz

ni

nt e n m n m 0 0 n n

n n i nnt e

For each given ξ , { tn(ξ ) } forms a representation of Td

1.3. Physical Consequences of Translational Symmetry

Let the simultaneous eigenstates of H & T(n) be | E k ( k = ξ / b )

H E k E k E k i k n bT n E k E k e

x-representation: Eku x x E k

Any x is related to a y in the unit cell by

x = n b + y –b/2 ≤ y < b/2 n

| x = T(n) | y x | = y | T †(n)

†Eku x y T n Ek y T n Ek i k nby Ek e i k x y

E ku y e

i k x i k yE k Eku x e u y e –b/2 ≤ y < b/2

i k xE k Ekv x u x e The Bloch function is periodic in x with period b

Reduced Schrodinger eq: 2 2 2 2

22 2Ek Ek

d kV y v y E v y

m d y m

Brillouin zone

… T(-1) T(0) T(1) …

ξ … exp(i ξ) 1 exp(-i ξ)

0 … 1 1 1

i nT n e

m\T … T(-1) T(0) T(1) …

0 … 1 1 1 …

1 … exp(i 2 π/N) 1 exp(i 2 π/N) …

1 … exp(i 2 π/N) 1 exp(i 2 π/N) …

2 /i n b k i n b m NbT nb k k e k e 2 m

kN

1.4. The Representation Functions and Fourier Analysis

Harmonic analysis: Functions as series of basic waves (harmonics)

E.g., Fourier series/transform, orthogonal polynomial expansion, …

i nn

n

f e f

*2

i n innn

de e

Orthonormality

*i n in

n

e e

Completeness

*

2i n

n

df e f

n nn

f

*n nd

*n n

n

*nd f Inverse

For compact groups, the basis of each group representation can be chosen to form a complete orthonormal set.

1.5. Symmetry Groups of Physics

Let G be the symmetry group of H • [ U(g),H ] = 0 g G Eigenstates of H are basis vectors of rep of G• Reps of G are independent of H• Some applications in which group theory is indispensable:

– Spectroscopy

– Electronic band theory

– Strong interaction / TOE

• Symmetry considerations can always simplify and / or clarify a problem.

E.g., expansion in terms of symmetrized functions

Some Symmetries in Physics

• Continuous Space-Time Symmetries– Translations in space– Translations in time– Rotations in space– Lorentz transformations

• Discrete Space-Time Symmetries– Space Inversion (Parity)– Time Reversal – Translations & rotations in a lattice (space groups)– Permutations of identical particles– Gauge invariance & charge conservation– Internal symetries of elementary particles