Lectures

Lecture 1: Second quantization formalism for harmonic oscillator

• Example: Coherent (quasiclassical) and squeezed states.
• Example: Isotropic three-dimensional harmonic oscillator.
• Example: Phonons in solids.

References

• Chapter 1.9 and 7.1 of Nazarov & Danon textbook.

Lecture 2: Second quantization formalism for bosons

• Example: Pair distribution function, density-density correlation function and structure factor.
• Example: Magnons in ferromagnets and antiferromagnets.
• Example: Bogoliubov theory of superfluidity.
• Example: Bose-Hubbard model for cold atoms in optical lattices.

References

• Chapters 3, 4.5 and 6 of Nazarov & Danon textbook.
• W. E. Lawrence, Algebraic identities relating first- and second-quantized operators, Am. J. Phys. 68, 167 (2000). [PDF]
• C. Timm, Lecture Notes on Theory of Magnetism
• J. M. Zhang and R. X. Dong, Exact diagonalization: The Bose–Hubbard model as an example, Eur. J. Phys. 31, 591 (2010). [PDF]

Lecture 3: Second quantization formalism for fermions

• Example: Hubbard dimer and trimer.
• Example: Mean-field magnetic phase diagram of Hubbard model.
• Example: Mean-field vs. exact Richardson solution of BCS model of superconductivity.
• Example: Hartree-Fock theory of electrons in metals.

References

• Chapters 3 and 5 of Nazarov & Danon textbook.
• C. Timm, Lecture Notes on Theory of Superconductivity
• D. J. Carrascal, J. Ferrer, J. C. Smith, and K. Burke, The Hubbard dimer: A density functional case study of a many-body problem, J. Phys.: Condens. Matter 29, 019501 (2017). [PDF]
• Y. Claveau, B. Arnaud and S. Di Matteo, Mean-field solution of the Hubbard model: the magnetic phase diagram, Eur. J. Phys. 35, 035023 (2014). [PDF]

Lecture 4: Time-dependent perturbation theory

• Example: Dyson vs. Magnus expansion for driven harmonic oscillator.

References

• Chapter 1 of Nazarov and Danon textbook.
• S. Blanes, F. Casas, J. A. Oteo, and J. Ros, A pedagogical approach to the Magnus expansion, Eur. J. Phys. 31, 907 (2010). [PDF]

Lecture 5: Floquet theory of periodically driven quantum systems

• Example: AC Stark effect for two-level atom in classical electromagnetic wave.
• Example: Floquet topological insulators.

References

• A. Eckardt and E. Anisimovas, High-frequency approximation for periodically driven quantum systems from a Floquet-space perspective, New J. Phys. 17, 093039 (2015). [PDF].

References

• B. Holstein, The adiabatic theorem and Berry’s phase, Am. J. Phys. 57, 1079 (1989). [PDF]
• V. T. Lahtinen and J. K. Pachos, A short introduction to topological quantum computation, SciPost Phys. 3, 021 (2017). [PDF]

Lecture 6: Quantization of the electromagnetic field

• Example: AC Stark effect for two-level atom in quantized electromagnetic field vs. Floquet theory.
• Example: Nonclassical light and photon statistics.
• Example: Light-matter interaction.

References

• M. Haas, U. D. Jentschura, and C. H. Keitel, Comparison of classical and second quantized description of the dynamic Stark shift, Am. J. Phys. 74, 77 (2006). [PDF]

Lecture 7: Dissipative quantum mechanics with application to qubits

• Example: Damped harmonic oscillator.
• Example: Qubit coupled to dissipative environment (spin-boson model).

References

• Chapters 11 and 12 of Nazarov & Danon textbook.

Lecture 8: Berry phase for time-dependent quantum systems

• Example: Spin in magnetic field.
• Example: Topological quantum computing.

Lecture 9: Scattering theory

References

• Chapter 19 of Shankar textbook.

Lecture 10: Relativistic quantum mechanics

References

• Chapter 13 of Nazarov & Danon textbook and Chapter 20 of Shankar textbook.