Graduate Problem Map

Graduate problems should make assumptions explicit. They should ask not only for an answer, but for the conditions under which the answer is meaningful.

Route

Stage	Problem Family	Canonical Page	Graduate Emphasis
1	Toolkit readiness	Diagnostic Checklist	domains, spectra, distributions, convergence
2	Formalism review	Exercises and Problems	projectors, mixed states, commutators, postulates
3	Approximation methods	Anharmonic Oscillator Perturbation	expansion parameter and matrix elements
4	Scattering	Gaussian Potential Born	normalization convention and validity of Born approximation
5	Semiclassical reasoning	WKB Barrier Tunneling	turning points and exponential estimates
6	Composite systems	Partial Trace Exercises	local statistics and entanglement diagnostics
7	Identical particles and Fock space	Identical-Particle Exercises and Fock-Space Exercises	exchange, occupation numbers, second-quantized notation
8	Numerical validation	Benchmark Problems	convergence and error interpretation

Representative Problem IDs

ID	Problem Type	Required Assumption
QM-PROB-G101	Prove or apply a commutator identity	operator domains or finite-dimensional setting
QM-PROB-G201	Use degenerate perturbation theory	chosen degenerate subspace and perturbation order
QM-PROB-G301	Evaluate a Born amplitude	weak, short-range potential and normalization convention
QM-PROB-G401	Estimate WKB tunneling	slowly varying barrier away from turning points
QM-PROB-G501	Compute a reduced density operator	tensor-factor ordering and subsystem definition
QM-PROB-G601	Validate a numerical eigenvalue calculation	refinement variable and reference result

Graduate Standards

A graduate solution should state:

• the Hilbert space and domain assumptions;
• the normalization convention;
• whether the spectrum is discrete, continuous, or mixed;
• the approximation parameter and retained order;
• the role of symmetry or degeneracy;
• the physical observable extracted from the calculation;
• at least one limiting case or benchmark.

Common Overclaims

• Treating a perturbative result as exact because the first correction is compact.
• Treating a finite-dimensional example as proof of an infinite-dimensional domain-sensitive statement.
• Calling a numerical plot evidence without a convergence study.
• Treating an effective model as fundamental without stating its regime.

References

• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, Wiley, 1977.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press, 1980.