Lecture Notes in Mathematical Quantum Mechanics

Lecture notes are useful when a compact proof sequence or a second set of examples would clarify a textbook argument. Their conventions and completeness vary, so the version and intended audience matter. The sources below are linked through author or institutional pages and are chosen for specific parts of the mathematical quantum mechanics route.

Lecture notes for unbounded operators

Jan Dereziński’s Unbounded Linear Operators, June 2013 version, develops closed operators, semigroups, Hilbert-space adjoints, forms, extensions, and concrete differential-operator models. The author’s notes directory identifies this as the course given at the Institut Henri Poincaré in June 2013.

Use the sections on closed operators and graph topology alongside Closed and Closable Operators and Tracking Domains. Then use the interval and half-line momentum examples to compare boundary conditions. The notes’ term “essential domain” should be translated through its definition: it is a graph-norm core, not merely a dense subset of Hilbert space.

A productive reading task is to follow one operator from its initial domain through its closure and adjoint. Record each domain separately. This prevents a familiar differential expression from concealing a change of realization.

Lecture notes for many-body analysis

Jan Philip Solovej’s Many Body Quantum Mechanics, notes updated 5 March 2014, are linked from the ESI course page. The course material connects many-body state spaces and spectral estimates to second quantization; the page also links supplementary notes on the min–max theorem and the Lieb–Thirring inequality.

Read these after the tensor-product and basic self-adjointness chapters. They are especially useful for moving from a formal many-particle Hamiltonian to questions about lower bounds and operator domains. Pair the notes with Symmetric and Antisymmetric Tensor Products before using occupation-number notation.

For spectral bounds, distinguish a variational upper bound from a theorem locating the whole essential spectrum. A successful trial-state calculation can be valuable without proving existence of an isolated eigenvalue. Keep the theorem’s threshold and compactness assumptions visible while reading.

Lecture notes for CCR and CAR representations

Dereziński’s “Introduction to representations of the canonical commutation and anticommutation relations” appeared in Large Coulomb Systems, Lecture Notes in Physics 695, 63–143 (2006). This is a focused route through representation-theoretic language, Fock constructions, and the distinction between bosonic and fermionic relations.

Use it with Weyl Form of the CCR, Regularity Assumptions, and Inequivalent CCR Representations. Check the symplectic normalization and continuity assumptions before comparing formulas with another source.

The central reading question is which conclusions are special to finitely many degrees of freedom and which survive in field or thermodynamic settings. A formal commutator and a regular Weyl representation are different mathematical inputs; the notes are useful precisely because representations are treated as objects in their own right.

Lecture notes for states, measurements, and channels

John Preskill’s Caltech notes provide an accessible companion for the finite-dimensional operational side. The course page identifies Chapter 2, Foundations I: States and Ensembles, and Chapter 3, Foundations II: Measurement and Evolution, as July 2015 versions.

Use Chapter 2 for ensemble and purification reasoning and Chapter 3 for operations and channel evolution. Pair them with Purification in Infinite Dimensions and Kraus Representation Limits when the target problem has infinitely many levels.

These notes are good sources of physical interpretation and finite-dimensional calculations. Their matrix arguments should not be used to infer trace-class convergence, unbounded-generator domains, or unrestricted entropy continuity without the additional infinite-dimensional statements.

Turn a note into a checked mathematical dependency

Choose one proposition needed for the current calculation. Write its full hypotheses in your own notation, identify whether its proof is supplied or cited, and test it on a simple model. If the proof uses a theorem not yet available, record that dependency instead of replacing it by “standard.”

Version labels also matter. An author’s current course page may link several generations of a chapter. Cite the title and version actually read, rather than combining one version’s page numbers with another version’s statement. Stable theorem numbers and named results are often more useful than PDF page counts.

The Proof Exercises page provides original short arguments for practicing this habit. The Textbooks guide identifies longer references when a note compresses a prerequisite too far.

Exercise: reconcile two uses of a domain word

One set of notes calls $\mathcal D$ an “essential domain” for a closed operator $A$, while another calls it a “core.” Both require approximation in $\|\psi\|+\|A\psi\|$. Are the objects different? Would ordinary norm density prove either condition?

Solution

With those explicit definitions they are the same object. The terminology differs, but the graph topology agrees. Ordinary norm density does not suffice because it controls vectors without controlling their images under $A$. A valid translation should preserve the defining convergence condition, as in Notation Translations.

References

• Dereziński, Jan. Unbounded Linear Operators. Lecture notes, Institut Henri Poincaré course, June 2013 version. Author’s PDF.
• Dereziński, Jan. “Introduction to representations of the canonical commutation and anticommutation relations.” In Large Coulomb Systems, Lecture Notes in Physics 695, Springer, 2006, pp. 63–143. doi:10.1007/3-540-32579-4_3; preprint.
• Preskill, John. Lecture Notes for Physics 219/Computer Science 219: Quantum Computation, Chapters 2 and 3, July 2015 versions. California Institute of Technology. Course and version index.
• Solovej, Jan Philip. Many Body Quantum Mechanics. Lecture notes, version updated 5 March 2014, ESI course. Institutional course page.