Prof. Enrico Bertuzzo

Quantum Mechanics I

General Infos

Purpose of the course: Quantum Mechanics is one of the fundamental subjects for any physicist (together with electromagnetism). In this course we will first introduce the operator and path integral formulation of QM (and show their equivalence), to later move on to applications of the formalism in various context. At the end of the course, the student is expected to be able to solve problem in 1 and 3 dimensions, to apply the raising and lowering operators to solve harmonic oscillator and angular momentum problems, and to have a clear idea of the equivalence between the operator and path integral formulations of the theory.

Lectures: Monday, Wednesday and Thursday (14h-16h), sala 2009 .

Evaluation: three written examinations (with both theoretical questions and exercises);

First midterm: Monday, September 9th;
Second midterm: Wednesday, October 23rd.
Third midterm: Wednesday, November 27th.

The grades will be assigned as follows:
A = 8.5 ≤ grade ≤ 10
B = 7 ≤ grade < 8.5
C = 5.5 ≤ grade < 7

NEW: FINAL GRADES (undergrad students)
NEW: FINAL GRADES (grad students)


Below you can find the preliminary plan for the lectures. It is possible that some details will change along the semester, but not the overall plan.

Books: D = Dirac, S = Sakurai, LB = Le Bellac, G = Gottfried-Yan, F = Feynman-Hibbs, M = Manousakis
  1. Mathematical structure of QM in finite dimensions 1: Hilbert space, dual space, self-adjoint operators and their properties [D 1, 2; LB 2.1, 2.2; S 1.2, 1.3; G 2.1]
  2. Mathematical structure of QM in finite dimensions 2: spectral decomposition of hermitian and unitary operators [D 2; LB 2.3]
  3. Postulates of QM [D 1,2; LB 4.1, 4.2; G 2.4]
  4. Applications of the postulates 1: ammonia and benzene molecule [LB 5.1]
  5. Applications of the postulates 2: time-independent perturbation theory [G 3.7a, 3.7b]
  6. Applications of the postulates 3: time-dependent problems and perturbation theory [M 22]
  7. The path integral formulation: construction of the path integral, classical limit, the wave function, Schrodinger equation, the path integral for the free particle [F 1, 2, 3, 4]
  8. Position and momentum operators
  9. Schrödinger equation for systems in d=1: general properties and simple examples
  10. Schrödinger equation for systems in d=1: harmonic oscillator (Frobenius and algebraic resolution)
  11. Schrödinger equation for systems in d=1: harmonic oscillator (path integral resolution and coherent states)
  12. Quantum Mechanics in 3 dimensions, Ehrenfest theorem, parity as a first example of symmetry
  13. Symmetries I: canonical quantization, symmetries from canonical quantization, action of symmetries on states, Wigner theorem, projective representations, Lie groups and Lie algebras
  14. Symmetries II: space and time translations, symmetries of the Hamiltonian, rotations in N dimensions, structure constants, rotations in 3 dimensions
  15. Symmetries III: construction of the unitary representations of SO(3), examples in 2 and 3 dimensions
  16. Symmetries IV: more on projective representations, SU(2) as the covering group of SO(3)
  17. Sum of angular momenta, Clebsch-Gordan coefficients [G3.5, LB10.6.2], and connection with group representations
  18. Orbital angular momentum and spherical harmonics
  19. Schrödinger equation for systems in d=3: free particle in spherical coordinates [G3.4], two body problem [G3.6]
  20. Schrödinger equation for systems in d=3: central potential and radial equation [G3.6b, LB10.4.1], hydrogen atom [G3.6c, LB10.4.2]
  21. Wigner-Eckart theorem and spherical tensors [see Robert Littlejohn lectures]
  22. ...


  1. List 1 (Lectures 2 and 3)
  2. List 2 (Lectures 4 and 5) updated 2019
  3. List 3 (Lectures 6 and 7) updated 2019
  4. List 4 (Lectures 8 and 9 plus more exercises on perturbation theory in finite dimensions
  5. List 5 (Position and momentum representations of QM)
  6. List 6 (Systems in 1 dimension and harmonic oscillator)
  7. List 7 (Symmetries) updated 2019
  8. List 8 (Orbital angular momentum and spin)
  9. List 9 (Hydrogen atom and angular momenta composition)
  10. List 10 (Wigner-Eckart, fine and hyperfine structure)

Recommended textbooks

It may also be useful to consult the following online references: