Group Theory

Question 1

Axioms of Quantum Mechanics

Answer 1

2

Question 2

Time evolution op.

Answer 2

14

Question 3

General group theory: definition of groups, multiplication tables, representations
with some examples (Z3, S3)

Answer 3

17-28 & 35-38

Question 4

Symmetries in quantum mechanics, relation to groups, example with Z2,
action of groups on functions

Answer 4

29-34

Question 5

Schur’s Lemma, example of applications (eg. dimension of irreps of Abelian
groups, simplified Wigner-Eckart theorem for scalar operators)

Answer 5

39-49

Question 6

Symmetries and conserved observables, exponential form of representations,
generators and Lie groups

Answer 6

50-56

Question 7

Irreps of SO(2) and U(1), example of the one dimensional harmonic oscillator

Answer 7

57-62

Question 8

Representations of SO(3) and its generators, structure constants, algebras
of so(n) and o(n)

Answer 8

63-71

Question 9

Representation of algebras, adjoint representation, Jacobi identity and
algebra of su(2)

Answer 9

72-77

Question 10

Relation between SU(2) and SO(3), double covers, scalar-spinor-vector
representations and sl(2,C)

Answer 10

78-82

Question 11

Irreps of su(2), maximum weight construction, matrix elements of generators
in the standard basis, Wigner D and d matrices

Answer 11

83-99

Question 12

Infinite dimensional representations of so(3), spherical harmonics and partial
wave expansions

Answer 12

100

Question 13

Representations on tensor product spaces, Clebsch-Gordan coefficients

Question 14

Spherical tensor operators, example with a vector and Wigner-Eckart theorem

Question 15

Casimir operators, space-time transformations and time-reversal