Further Reading
Choose a source for the calculation you want to understand. A first pass through operator evolution needs different preparation from a proof about an unbounded Hamiltonian or a field-theory correlation function. The selections below give a starting point and a concrete next calculation; full publication details appear at the end.
Reading Quantum Dynamics by Problem
Section titled “Reading Quantum Dynamics by Problem”| Question | Start with | Calculation to try next |
|---|---|---|
| How do states and operators evolve? | Shankar for a gradual introduction; Sakurai and Napolitano for a compact graduate treatment | Derive spin precession in two pictures using picture translations |
| When does a formal Hamiltonian define unitary motion? | Teschl, Chapters 2–5 | Explain why a boundary condition belongs to the operator in Hamiltonians as Generators |
| How does a path sum reproduce a kernel? | Schulman; Grosche’s freely available lecture notes for worked examples | Recover the normalization as well as the phase of the free kernel |
| What does a resolvent reveal about a spectrum? | Economou | Derive the sign of the imaginary part in the density of states |
| How can quantum dynamics be written in phase space? | Curtright and Zachos for an overview; Schleich for optical states | Derive both marginals of the Wigner function before computing its evolution |
| How should a driven calculation be organized? | Tannor; Bukov, D’Alessio, and Polkovnikov for periodic drives | Compare exact propagation and a high-frequency approximation in Floquet theory |
| Why do fields use time-ordered correlators? | Tong’s freely available QFT lectures, Sections 2–3 | Compare a particle kernel with a field propagator |
The mathematical route requires familiarity with Hilbert spaces and elementary real analysis. Teschl’s preliminary material supplies the functional-analysis vocabulary; its quantum-dynamics chapter treats Stone’s theorem and the Trotter formula. A physical calculation can be learned first, but its domain assumptions should be revisited when moving from matrices to differential operators.
A First Route Through the Subject
Section titled “A First Route Through the Subject”Work through a two-level system before taking up continuous kernels: it separates ordering and picture changes from integrals and boundary conditions. Next compute a free Gaussian packet using spectral evolution and a propagator. The agreement makes the normalization of the kernel testable. Then derive the same kernel by time slicing, and only afterward introduce sources or a field with infinitely many modes.
For phase-space work, start with one Gaussian state whose position and momentum widths are known. A correct transform must reproduce those marginals and the uncertainty relation. Non-Gaussian states become much easier to interpret after this check.
When comparing references, identify the object before translating its notation. A time-evolution kernel, an energy resolvent, and a time-ordered field correlator can all be called a propagator. The Green Function Table, Path Integral Conventions, and Phase-Space Conventions state the definitions used here.
References
Section titled “References”- M. Bukov, L. D’Alessio, and A. Polkovnikov, “Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to Floquet engineering,” Advances in Physics 64, 139–226 (2015). Author preprint.
- T. L. Curtright and C. K. Zachos, “Quantum mechanics in phase space,” Asia Pacific Physics Newsletter 1, 37–46 (2012). Author preprint.
- E. N. Economou, Green’s Functions in Quantum Physics, 3rd ed., Springer (2006).
- C. Grosche, “An introduction into the Feynman path integral,” lecture notes (1993). Author preprint.
- J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press (2020).
- W. P. Schleich, Quantum Optics in Phase Space, Wiley-VCH (2001).
- L. S. Schulman, Techniques and Applications of Path Integration, Wiley (1981).
- R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer (1994).
- D. J. Tannor, Introduction to Quantum Mechanics: A Time-Dependent Perspective, University Science Books (2007).
- G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., Graduate Studies in Mathematics 157, American Mathematical Society (2014). Author’s page with an authorized online edition.
- D. Tong, Quantum Field Theory, University of Cambridge lecture notes. Author’s course page.