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: Magnons in ferromagnets and antiferromagnets.
• Example: Bogoliubov theory of superfluidity.

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]

Lecture 3: Second quantization formalism for fermions

• Example: Exact diagonalization of Hubbard clusters.
• Example: BCS theory of superconductivity.
• Example: Hartree-Fock theory of electrons in metals.

References

• Chapters 3 and 5 of Nazarov & Danon textbook.

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: Periodically driven harmonic oscillator.
• 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].

Lecture 6: Berry phase for time-dependent quantum systems

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

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 7: Quantization of the electromagnetic field

• Example: Casimir effect.
• Example: Nonclassical light and photon statistics.
• Example: Light-matter interaction.

Lecture 8: Dissipative quantum mechanics with application to qubits

• Example: Damped harmonic oscillator.
• Example: Qubit coupled to dissipative environment.

References

• Chapters 11 and 12 of Nazarov & Danon textbook.

Lecture 10: Relativistic quantum mechanics

References

• Chapter 13 of Nazarov & Danon textbook.