Researcher Quick Reference

This is a dense lookup sheet for readers who already know quantum mechanics and need to check notation, assumptions, canonical pages, formulas, or source quality quickly. It is useful when reading a paper, preparing a lecture, comparing conventions across subfields, or deciding which canonical page should carry a result.

For a guided repair route, use the Researcher Refresher Roadmap. For student-facing problem workflow, use the Student Quick Reference.

First Triage

Task	Fast route	Do not skip
Check notation	Common Convention Translations and Common Symbols Index	Units, Fourier signs, inner-product slot, tensor order.
Check a formula	Core Formulas Index	Assumptions and canonical-home link.
Identify a Hamiltonian	Most-Used Hamiltonians	Hilbert space, domain, gauge, and approximation.
Audit a calculation	Common Checks and Sanity Tests	Normalization, positivity, dimensions, limits.
Find a source	Bibliography and Reading Guides	Match source type to claim type.
Connect to fields or many-body language	QFT Bridge Index and Second Quantization	Fixed-particle assumptions and particle-number sectors.
Judge claim status	Scientific standards	Standard result, approximation, interpretation, active research, or speculation.

Structural Formulas

These are safe as lookup anchors only when their hypotheses are present.

Structure	Formula	Hypotheses to check
Pure-state probability	$P(P_a)=\langle\psi\vert P_a\vert\psi\rangle$	Normalized state; projector belongs to the measurement.
Density-operator expectation	$\langle A\rangle=\operatorname{Tr}(\rho A)$	$\rho$ positive trace one; trace exists.
Reduced state	$\rho_A=\operatorname{Tr}_B\rho_{AB}$	Tensor-product ordering and traced subsystem are declared.
Spectral decomposition	$A=\sum_a aP_a$	Discrete finite or controlled spectral case; otherwise use spectral measures.
Time-independent evolution	$U(t)=e^{-iHt/\hbar}$	$H$ self-adjoint and time independent.
Heisenberg equation	$dA_H/dt=(i/\hbar)[H,A_H]+\partial A_H/\partial t$	Domains and explicit time dependence are controlled.
Canonical commutator	$[\hat x,\hat p]=i\hbar$	Representation and domains matter; not an ordinary matrix identity in finite dimension.
Perturbative split	$H=H_0+\lambda V$	Solved $H_0$, dimensionless small parameter, degeneracy checked.
Variational bound	$E_0\le\langle\psi\vert H\vert\psi\rangle/\langle\psi\vert\psi\rangle$	Trial state lies in the Hamiltonian form domain.
Scattering cross section	$d\sigma/d\Omega=\lvert f(\theta,\phi)\rvert^2$	Normalization, asymptotic convention, and flux convention match.
Many-particle Hamiltonian	$H_N=\sum_\alpha h^{(\alpha)}+\sum_{\alpha<\beta}v^{(\alpha\beta)}$	Particle statistics, symmetry sector, and interaction convention are declared.

Canonical starts: Density Operators, Spectral Decomposition, Time-Evolution Operator, Canonical Commutation Relations, Approximation Decision Tree, Scattering Amplitude, and Many-Particle Hamiltonians.

Paper Reading Audit

Before importing a result from a paper or another subfield, identify:

1. Hilbert space, sector, and representation.
2. State language: pure states, density operators, path integrals, Green functions, or occupation numbers.
3. Hamiltonian, Lagrangian, generator, or effective model.
4. Units, constants, Fourier convention, and phase convention.
5. Boundary conditions, gauge choice, and self-adjoint domain when relevant.
6. Symmetries, conserved quantities, and selection rules.
7. Observable, measurement, correlation function, or response function being computed.
8. Approximation method, expansion parameter, cutoff, or limiting process.
9. Numerical discretization, convergence checks, or benchmark comparison.
10. Claim type: theorem, derivation, approximation, experimental result, numerical evidence, interpretation, open problem, or speculative idea.

If a paper is excellent but uses different conventions, cite it honestly and translate explicitly. If a result depends on a convention or approximation, the page should say so near the formula, not only in a reference list.

Rigor Flags

Flag	Why it matters	First route
Unbounded operator	Formal symmetry is not self-adjointness. Domains can change spectra.	Spectral Theorem: Practical Version
Continuous spectrum	Eigenkets are generalized objects; normalization is distributional.	Discrete and Continuous Spectra
Trace or entropy in infinite dimension	Trace-class and convergence assumptions are not automatic.	Trace-Class and Hilbert-Schmidt Operators
Degeneracy	Nondegenerate perturbation and naive eigenvector tracking can fail.	Degenerate Perturbation Theory
Gauge field	Canonical and kinetic momentum differ; phases may be gauge dependent.	Minimal Coupling
Identical particles	Symmetric or antisymmetric sectors are part of the state space.	Symmetrization Postulate
Thermodynamic or continuum limit	Order of limits can change the result.	Many-Particle Hamiltonians
Open-system dynamics	A map on density operators needs positivity or complete positivity assumptions.	Density Operators

When one of these flags appears, a compact reference entry is not enough. Follow the canonical page and cite a source whose level matches the issue.

Symmetry and Representation Checks

Use symmetry before calculation.

• If $[H,A]=0$, check whether $A$ gives a conserved quantity or block diagonalization.
• If a perturbation is odd under parity, diagonal matrix elements in parity eigenstates may vanish.
• If angular momentum is present, identify whether $L$, $S$, or $J$ is the relevant generator.
• If time reversal is present, check whether the symmetry is antiunitary.
• If a theorem assumes unitary symmetries, do not silently apply it to antiunitary symmetries.

Canonical starts: Quantum Symmetries, Unitary Symmetries, Antiunitary Symmetries, and Symmetry Constraints on Hamiltonians.

Approximation and Numerical Control

Do not record an approximation as a formula without its control statement.

Method	Control question
Nondegenerate perturbation theory	Are off-diagonal couplings small compared with relevant energy gaps?
Degenerate perturbation theory	Has the perturbation been diagonalized in the degenerate or nearly degenerate subspace?
Variational method	Is the trial family in the correct domain and does the bound direction matter?
WKB	Is the de Broglie wavelength slowly varying away from turning points?
Born approximation	Is the scattering potential weak in the relevant dimensionless sense?
Effective Hamiltonian	What subspace was retained and what scale was integrated out?
Numerical diagonalization	Are basis size, grid spacing, boundary conditions, and benchmark cases checked?

Useful starts: Approximation Decision Tree, Small Parameters and Error Estimates, First Born Approximation, and Phase Shifts.

Canonical-Home Discipline

The Reference should usually be the canonical index, not the canonical derivation. When adding or editing a research-facing entry:

1. Link to the canonical derivation instead of repeating it.
2. State the assumptions and convention in the entry itself.
3. Cite a source appropriate to the claim.
4. Mark frontier claims with evidence labels.
5. Do not upgrade a conjectural or interpretive statement into settled formalism.
6. Add a changelog entry only for reader-facing corrections, convention changes, major reorganizations, or formula-level fixes.

Use the scientific standards when checking reasoning and sources, and contact the maintainer to suggest an improvement.

Source Fit

Claim type	Strong source type
Standard derivation	Graduate textbook or monograph.
Operator-domain theorem	Mathematical physics monograph or rigorous text.
Historical priority	Original paper plus reliable secondary history.
Active research result	Recent review and primary papers.
Experimental claim	Experimental paper, collaboration paper, or review with data context.
Software behavior	Official documentation and method paper when behavior affects results.
Convention comparison	The source whose convention is being translated plus the canonical convention page.

Start with Textbooks and Classic Papers, then add topic-specific sources as the Reference expands.

Common Researcher Pitfalls

• Recognizing a formula but not its convention.
• Treating a finite-dimensional proof as valid for unbounded operators.
• Importing a result from a different normalization or scattering convention.
• Forgetting that a density matrix representation is basis dependent.
• Treating a numerical plot as evidence without convergence or benchmark checks.
• Letting a paper’s interpretive framing replace the operational content of the formalism.
• Turning a compact reference entry into a duplicate derivation.
• Citing a source for authority without checking that its assumptions match the page.

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press, 1980.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.
• A. Peres, Quantum Theory: Concepts and Methods, Kluwer, 1995.

Exercises

1. A paper states $[\hat x,\hat p]=i$ and $U(t)=e^{-iHt}$. What should a reference entry do before citing its formula?

Solution

Identify that the paper is using $\hbar=1$, translate the formulas to the default explicit-$\hbar$ convention, and state the translation near the formula if the cited result is imported.

2. A derivation uses a formally Hermitian differential operator but never states boundary conditions. What is the research-level warning?

Solution

The operator may not yet be self-adjoint. Boundary conditions and domains can change the spectrum and whether the Hamiltonian generates unitary evolution. The page should either state the domain assumptions or link to a source that does.

3. A compact reference entry starts to reproduce a four-page perturbation derivation already present elsewhere. What should happen?

Solution

Keep the reference entry compact: state the result, assumptions, and warning, then link to the canonical derivation. Repeating the derivation risks stale duplicated content.