Quantum Mechanics I — Course Plan (2026.1)
Graduate Program in Physics — UERJ Prof. Marcelo Santos Guimaraes
Period: 09/03/2026 - 10/07/2026 Schedule: Mondays and Wednesdays, 1h40 per class; final assessment closeout on Friday, 10/07 Total workload: 60 hours (36 regular meetings x 1h40; final assessment block closed by 10/07)
Course overview
This course presents quantum mechanics at the graduate level, balancing formal rigor and physical intuition, with a focus on quantum information. The structure follows the logic:
formalism -> concrete applications -> quantum information perspective
The material is organized into six thematic modules, interleaved with five in-person assessments. Modules I-III closely follow the instructor’s lecture notes; Modules IV-VI complement the course with symmetries, approximation methods, and scattering.
Main references
- [Sakurai] Sakurai, J.J. & Napolitano, J. — Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2021.
- [NC] Nielsen, M.A. & Chuang, I.L. — Quantum Computation and Quantum Information, Cambridge University Press, 2000.
- [Ballentine] Ballentine, L.E. — Quantum Mechanics: A Modern Development, 2nd ed., World Scientific, 2015.
Additional references:
- [CT] Cohen-Tannoudji, C., Diu, B. & Laloë, F. — Quantum Mechanics, vols. 1 and 2, Wiley-VCH, 2020 (3rd ed.).
- [Wilde] Wilde, M.M. — Quantum Information Theory, 2nd ed., Cambridge University Press, 2017.
- [Preskill] Preskill, J. — Lecture Notes on Quantum Computation, available at https://www.preskill.caltech.edu/ph219/.
Assessment structure
Assessment will be done through 5 problem sets. The sets will be assigned throughout the course and graded in in-person sessions in which students solve selected problems on the board. In the adjusted calendar, each assessment block uses two meetings. The final teaching activity closes on 10/07/2026, and grades should be ready by 24/07/2026.
| Assessment | Dates | Content |
|---|---|---|
| Set 1 | 06/04 and 08/04/2026 | Module I — Principles and formalism |
| Set 2 | 27/04 and 29/04/2026 | Module II — Quantum dynamics |
| Set 3 | 20/05 and 25/05/2026 | Module III — Open systems and quantum information |
| Set 4 | 10/06 and 15/06/2026 | Module IV — Symmetries and angular momentum |
| Set 5 | 08/07 and 10/07/2026 | Modules V and VI — Approximation and scattering |
Detailed schedule
Module I — Principles and Formalism (8 classes)
Main references: Lecture notes Sec. 1-1.5; [Sakurai] Chs. 1-2; [Ballentine] Chs. 1-3; [NC] Sec. 2.1-2.2.
| # | Date | Day | Topic |
|---|---|---|---|
| 1 | 09/03 | Mon | Course introduction. Axioms 1 and 2. Hilbert space; inner product; Dirac notation (kets, bras). Observables as self-adjoint operators; spectral representation; orthogonal projections. |
| 2 | 11/03 | Wed | Axioms 3, 4, and 5. Measurement: probabilities, Born rule, collapse. Dynamics: Schrodinger equation, unitary time-evolution operator. Composite systems: tensor product, first notion of entanglement. |
| 3 | 16/03 | Mon | Operator algebra and representations. C*-algebras; spectrum and norm; von Neumann algebra. States as normalized positive linear forms. GNS construction (overview and physical meaning). |
| 4 | 18/03 | Wed | Finite-dimensional Hilbert spaces. Matrix representations; basis changes; isometries. Degeneracy; complete set of commuting observables (CSCO). |
| 5 | 23/03 | Mon | Uncertainty relations. Infinite-dimensional spaces. Robertson inequality. Position and momentum operators; canonical commutation relation; Dirac delta; rigged Hilbert space. |
| 6 | 25/03 | Wed | Harmonic oscillator. Algebraic method: creation and annihilation operators; spectrum; wavefunctions as Hermite polynomials. Number-angle variables. |
| 7 | 30/03 | Mon | Coherent states. Eigenstates of the annihilation operator; Poisson distribution; completeness properties (overcompleteness). Displacement operator D(z). |
| 8 | 01/04 | Wed | Squeezed states. Squeezing operator S(r); Bogoliubov transformations; minimum uncertainty in phase space. Graphical representation in phase space. |
Note: Class 8 closes Module I. Set 1 should be distributed around class 5 so students have time to prepare.
| # | Date | Day | Activity |
|---|---|---|---|
| A1.1 | 06/04 | Mon | Assessment 1 — Set 1 (Module I), day 1 |
| A1.2 | 08/04 | Wed | Assessment 1 — Set 1 (Module I), day 2 |
Module II — Quantum Dynamics (4 classes)
Main references: Lecture notes Sec. 1.6-1.7; [Sakurai] Chs. 2 and 15; [Ballentine] Ch. 4; [CT] vol. 1, appendix H_III.
| # | Date | Day | Topic |
|---|---|---|---|
| 9 | 13/04 | Mon | Time evolution. Evolution operator; Schrodinger equation for states and operators. Neutrino oscillations as a paradigmatic example of two-level systems. |
| 10 | 15/04 | Wed | Schrodinger and Heisenberg pictures. Heisenberg equation; time-energy uncertainty relation. Harmonic oscillator dynamics in both pictures. |
| 11 | 20/04 | Mon | Motion of coherent states. Evolution under a general time-dependent Hamiltonian (driven oscillator). Connection with classical dynamics. |
| 12 | 22/04 | Wed | Propagator and path integral. Feynman kernel; derivation for the free particle and harmonic oscillator. Sum-over-histories formulation (overview). Potentials, gauge transformations, and the Aharonov-Bohm effect. |
Note: Set 2 should be distributed around class 10.
| # | Date | Day | Activity |
|---|---|---|---|
| A2.1 | 27/04 | Mon | Assessment 2 — Set 2 (Module II), day 1 |
| A2.2 | 29/04 | Wed | Assessment 2 — Set 2 (Module II), day 2 |
Module III — Open Quantum Systems and Quantum Information (5 classes)
Main references: Lecture notes Sec. 2; [NC] Chs. 2, 4, and 8; [Sakurai] Ch. 3 (density operators); [Wilde] Chs. 3-4; [Preskill] Ch. 2.
| # | Date | Day | Topic |
|---|---|---|---|
| 13 | 04/05 | Mon | Composite systems and the density operator. Tensor product revisited; entanglement. Density operator: axiomatic definition, pure and mixed states. Partial trace and reduced state. |
| 14 | 06/05 | Wed | Qubits. Bloch sphere; Pauli matrices as a basis for operators in dimension 2. Completely depolarized state. Purification of mixed states. |
| 15 | 11/05 | Mon | Entanglement and nonlocality. Bell states; Bell basis. CHSH inequality. Quantum violation and Bell’s theorem. Connection with QM foundations. |
| 16 | 13/05 | Wed | Von Neumann entropy. Definition; properties: additivity, subadditivity, invariance under unitary evolution. Entanglement entropy as an entanglement measure. Von Neumann equation (quantum analog of Liouville). |
| 17 | 18/05 | Mon | Quantum channels and QI protocols. Evolution of open systems; Kraus operators; quantum channel as a completely positive trace-preserving (CPTP) map. No-cloning theorem. Quantum teleportation and superdense coding. |
Note: Set 3 should be distributed around class 15. The assessment starts after the entropy and channels classes.
| # | Date | Day | Activity |
|---|---|---|---|
| A3.1 | 20/05 | Wed | Assessment 3 — Set 3 (Module III), day 1 |
| A3.2 | 25/05 | Mon | Assessment 3 — Set 3 (Module III), day 2 |
Module IV — Symmetries and Angular Momentum (4 compressed classes)
Main references: Lecture notes Sec. 3-4; [Sakurai] Chs. 3-4; [Ballentine] Chs. 3 and 7; [CT] vol. 1, Chs. VI-VII and supplements.
| # | Date | Day | Topic |
|---|---|---|---|
| 18 | 27/05 | Wed | Symmetries and spatial transformations. Unitary and anti-unitary transformations (Wigner theorem). Spatial translations and rotations; generators; angular momentum algebra. |
| 19 | 01/06 | Mon | Spin and angular-momentum representations. Raising and lowering operators; quantization of j and m. Irreducible representations of SU(2); spin-1/2 and Pauli matrices. |
| 20 | 03/06 | Wed | Orbital angular momentum. Spherical harmonics; Schrodinger equation in spherical coordinates. Relation between SO(3), SU(2), and orbital representations. |
| 21 | 08/06 | Mon | Addition of angular momenta and discrete symmetries. Clebsch-Gordan coefficients; spin-orbit coupling; Wigner-Eckart theorem (statement). Parity and time reversal. |
Note: Set 4 should be distributed around class 20. This version keeps the core representation theory and symmetry tools, while moving detailed applications into the problem set and review.
| # | Date | Day | Activity |
|---|---|---|---|
| A4.1 | 10/06 | Wed | Assessment 4 — Set 4 (Module IV), day 1 |
| A4.2 | 15/06 | Mon | Assessment 4 — Set 4 (Module IV), day 2 |
Modules V and VI — Approximation Methods and Scattering Theory (6 compressed classes)
Main references: [Sakurai] Chs. 5-7; [CT] vol. 2, Chs. VIII and XI-XIII; [Ballentine] Chs. 11-13.
| # | Date | Day | Topic |
|---|---|---|---|
| 22 | 17/06 | Wed | Time-independent perturbation theory. Non-degenerate and degenerate cases; perturbative corrections to energies and states; fine structure as an organizing example. |
| 23 | 22/06 | Mon | Zeeman effect and variational method. Normal and anomalous Zeeman effects. Rayleigh-Ritz variational principle; application to helium. Connection with QI: variational eigenvalue estimation as an optimization problem. |
| 24 | 24/06 | Wed | Time-dependent perturbation theory and Fermi’s golden rule. Interaction picture; Dyson series; transitions between two levels; transitions to a continuum. |
| 25 | 29/06 | Mon | General formulation of scattering. Lippmann-Schwinger equation; scattering states; differential cross section. Born approximation as the first calculational tool. |
| 26 | 01/07 | Wed | Born approximation and partial waves. First Born approximation; Yukawa and Coulomb examples; partial-wave decomposition; phase shifts. |
| 27 | 06/07 | Mon | Optical theorem and synthesis. Total cross section; optical theorem; overview of the S-matrix and unitarity as an information-conservation principle. |
Note: Set 5 should be distributed around class 24. The final assessment block uses Wednesday 08/07 and Friday 10/07, the last allowed teaching day.
| # | Date | Day | Activity |
|---|---|---|---|
| A5.1 | 08/07 | Wed | Assessment 5 — Set 5 (Modules V and VI), day 1 |
| A5.2 | 10/07 | Fri | Assessment 5 — Set 5 (Modules V and VI), day 2 |
Calendar summary
| Week | Monday | Wednesday |
|---|---|---|
| 1 (09-13/03) | Class 1 — Axioms 1-2 | Class 2 — Axioms 3-5 |
| 2 (16-20/03) | Class 3 — C*-algebras, GNS | Class 4 — Finite dim., CSCO |
| 3 (23-27/03) | Class 5 — Uncertainty, infinite dim. | Class 6 — Harmonic osc. |
| 4 (30/03-03/04) | Class 7 — Coherent states | Class 8 — Squeezed states |
| 5 (06-10/04) | Assessment 1, day 1 | Assessment 1, day 2 |
| 6 (13-17/04) | Class 9 — Time evolution | Class 10 — Schrodinger/Heisenberg |
| 7 (20-24/04) | Class 11 — Coherent states: dynamics | Class 12 — Propagator, gauge, A-B |
| 8 (27/04-01/05) | Assessment 2, day 1 | Assessment 2, day 2 |
| 9 (04-08/05) | Class 13 — rho-hat, partial trace | Class 14 — Qubits, Bloch |
| 10 (11-15/05) | Class 15 — Bell, CHSH | Class 16 — Von Neumann entropy |
| 11 (18-22/05) | Class 17 — Channels, teleportation | Assessment 3, day 1 |
| 12 (25-29/05) | Assessment 3, day 2 | Class 18 — Symmetries |
| 13 (01-05/06) | Class 19 — Spin and angular momentum | Class 20 — Orbital ang. mom. |
| 14 (08-12/06) | Class 21 — Addition and discrete sym. | Assessment 4, day 1 |
| 15 (15-19/06) | Assessment 4, day 2 | Class 22 — Perturbation theory |
| 16 (22-26/06) | Class 23 — Zeeman, variational | Class 24 — Time-dep. perturbation |
| 17 (29/06-03/07) | Class 25 — Scattering formulation | Class 26 — Born and partial waves |
| 18 (06-10/07) | Class 27 — Optical theorem, synthesis | Assessment 5, day 1 |
Final class limit: Friday 10/07 is reserved for Assessment 5, day 2. Grades should be ready by 24/07.
Problem set distribution
| Set | Suggested assignment | Assessment | Covered topics |
|---|---|---|---|
| 1 | Class 5 (23/03) | 06/04 and 08/04 | Hilbert space; Dirac notation; spectral representation; matrix representations; uncertainty relations; harmonic oscillator; coherent and squeezed states. |
| 2 | Class 10 (15/04) | 27/04 and 29/04 | Time evolution; Schrodinger and Heisenberg pictures; two-level oscillations; oscillator dynamics; propagator; Aharonov-Bohm effect. |
| 3 | Class 15 (11/05) | 20/05 and 25/05 | Density operator; partial trace; qubits and Bloch sphere; entanglement; Bell inequality; von Neumann entropy; quantum channels; teleportation. |
| 4 | Class 20 (03/06) | 10/06 and 15/06 | Translations and rotations; angular momentum representations; spherical harmonics; addition of angular momenta; Clebsch-Gordan; parity; time reversal. |
| 5 | Class 24 (24/06) | 08/07 and 10/07 | Time-independent and time-dependent perturbation theory; fine structure; Zeeman effect; variational method; Fermi’s golden rule; scattering; Lippmann-Schwinger; Born; partial waves. |
Quantum information connections throughout the course
One of the distinguishing features of this course is the systematic integration of quantum information concepts. The table below summarizes the planned connections in each module:
| Module | QI connection |
|---|---|
| I — Formalism | Qubits as dimension-2 systems; states as resources; computational basis {|0⟩, |1⟩} and its identification with spin-1/2. |
| II — Dynamics | Quantum gates as unitary operators; controlled evolution; geometric phase and holonomic quantum computing (brief mention). |
| III — Open systems and QI | Core QI module: Bell states, CHSH, teleportation, superdense coding, quantum channels, no-cloning. |
| IV — Symmetries | Symmetries as quantum gates (Pauli, Hadamard, CNOT via spin); entanglement and representations; Wigner theorem and allowed operations. |
| V — Approximation | Quantum Hamiltonian simulation; variational problem as quantum optimization (VQE - mention); perturbation as an error model. |
| VI — Scattering | S-matrix and unitarity as information conservation; quantum channel and scattering theory perspective. |