Diagnostic Checklist

Use this checklist to decide what to review before reading a quantum-mechanics page. It is not an exam. It is a triage tool: a few missed items tell you where to slow down, which Toolkit page to open, and which convention to check before a calculation goes sideways.

The most useful habit is to test yourself with small computations. If you can explain the words but cannot do a two-line example, treat the topic as worth reviewing.

Linear Algebra Readiness

You are ready for the first Core Formalism pages if you can:

write a vector as a linear combination of basis vectors;
compute inner products of complex vectors using conjugation in the first slot;
normalize a nonzero vector;
find eigenvalues and eigenvectors of a small matrix;
recognize Hermitian, unitary, and projector matrices;
use a spectral decomposition for a finite Hermitian matrix;
multiply simple tensor-product states and operators.
identify rank and null directions from singular values.
recognize Schmidt coefficients as singular values of a bipartite coefficient matrix.
translate a one-qubit density matrix into Pauli-coordinate Bloch-vector form.

Repair links:

Hilbert-Space Readiness

You are ready for wavefunction and spectrum pages if you can:

explain why square-integrability matters for a wavefunction;
distinguish a vector from its coordinates in a basis;
compute a norm such as $\lVert\psi\rVert^2$ from an inner product;
distinguish pointwise convergence from Hilbert-space norm convergence;
state why continuous position kets are generalized objects;
distinguish Hilbert-space norm convergence from pointwise convergence;
explain why $L^2(\mathbb R)$ can be separable despite continuous position labels;
define a bounded operator and its operator norm;
recognize that unbounded operators require domain care, even when a first calculation suppresses it.
explain how an adjoint moves an operator across an inner product.
distinguish a symmetric operator from a self-adjoint operator in domain-sensitive examples.
read a spectral projection $E_A(\Delta)$ as the probability projector for values in $\Delta$ .
treat continuous-spectrum eigenkets as generalized vectors rather than normalizable states.
explain that generalized position and momentum kets live on the distribution side of a rigged Hilbert space.
know that infinite-dimensional density operators must be trace-class, not merely positive bounded operators.
distinguish normalizability from continuity, differentiability, and finite moments.

Repair links:

Fourier and Distribution Readiness

You are ready for position-momentum, wave-packet, and scattering pages if you can:

expand a periodic function in discrete Fourier modes;
read complex exponential phases as oscillations, plane waves, and Fourier modes;
describe Fourier transformation as a change between complementary descriptions;
track the normalization convention and the factors of $2\pi$ and $\hbar$ ;
use Plancherel or Parseval to check that a Fourier transform or Fourier series preserves probability normalization;
translate basic operators into momentum representation;
recognize when a Green-function or response formula is a convolution;
identify the reciprocal-lattice spacing in a Poisson summation formula;
compute the transform of a Gaussian up to convention-dependent constants;
use the delta function under an integral;
read a distributional identity as an equality after pairing with test functions;
compute the distributional derivative of a step or jump;
interpret principal-value singular integrals and $i0$ prescriptions;
explain why a sharply localized position state is not a normalizable wavefunction.
recognize analytic functions as the objects that allow contour deformation and analytic continuation.
recognize poles, branch cuts, and contour choices in advanced Fourier and scattering formulas.

Repair links:

Differential-Equation Readiness

You are ready for canonical wave-mechanics pages if you can:

solve a second-order ordinary differential equation with constant coefficients;
apply boundary conditions to determine allowed constants;
recognize an eigenvalue problem;
explain what a product ansatz and separation constant do.
explain how boundary conditions can change a spectrum;
know when a series solution may be differentiated or used only as a norm-convergent expansion.
distinguish initial data from boundary data in a partial differential equation.
read a Sturm–Liouville problem as an orthogonality and completeness structure.
recognize when special functions such as Hermite, Legendre, Laguerre, Bessel, Airy, and hypergeometric functions arise from weighted or asymptotic eigenfunction problems.
read a Green function as an inverse kernel with specified boundary conditions.
derive an Euler–Lagrange equation from a first variation.
connect a Lagrangian action to Euler–Lagrange equations and the path-integral phase.
translate a regular Lagrangian into a Hamiltonian and write Hamilton’s equations.
distinguish a classical phase-space point from a quantum Hilbert-space state.
compute simple Poisson brackets and relate $\{q,p\}=1$ to $[Q,P]=i\hbar I$ with caveats.
test whether a simple change of phase-space variables is canonical.
recognize the standard symplectic matrix $J$ and the condition $M^TJM=J$ .
explain that a symplectic manifold has a closed nondegenerate 2-form and Hamiltonian vector fields.
state the Poisson-to-commutator correspondence and name at least one ordering caveat.
recognize the Hamilton–Jacobi equation as the leading action-phase equation behind WKB.
distinguish stationary action from least action and identify which endpoint variations are fixed.
identify the dimensionless small parameter in a semiclassical approximation and explain why stationary phase selects classical paths.
recognize a functional derivative as the gradient of a functional with respect to a function.
distinguish an asymptotic expansion from a convergent series.

Repair links:

Probability and Information Readiness

You are ready for Born-rule, density-operator, and entropy pages if you can:

identify a sample space, event, probability measure, and random variable in a simple experiment;
normalize a discrete probability distribution;
normalize a probability density;
recognize the mean and variance conventions of a Gaussian density;
compute an expectation value and variance;
compute the Shannon entropy of a finite distribution;
recognize KL divergence as a comparison between two distributions;
identify Fisher information as local sensitivity of a parameterized model;
estimate a sample mean and its Monte Carlo standard error;
explain why incompatible quantum observables may lack one classical joint distribution;
distinguish an amplitude from a probability;
identify when probabilities are classical ignorance and when a density operator represents reduced quantum information.

Repair links:

Symmetry Readiness

You are ready for spin, angular momentum, and symmetry pages if you can:

check the group axioms in simple examples such as $\mathbb Z_N$ , $U(1)$ , and two-element parity groups.
recognize a group action as a transformation of states or coordinates;
distinguish discrete groups from Lie groups with continuous parameters and infinitesimal generators;
read a Lie algebra bracket as a commutator structure rather than ordinary multiplication;
distinguish unitary from antiunitary implementations and know why antiunitary maps conjugate $i$ ;
compute commutators of simple matrices;
use the Pauli matrices as a basis for qubit observables;
explain why $SU(2)$ is related to spin rotations;
use ladder operators in a finite angular-momentum multiplet.
recognize that Berry-phase paths live in parameter spaces, which may need more than one coordinate chart.
distinguish a tangent vector from a cotangent vector, and know why a differential is not automatically a gradient vector.
read a 1-form as something integrated along a curve and a 2-form as something integrated over a surface.
know that the exterior derivative raises form degree and satisfies $d^2=0$ .
use Stokes theorem with the correct form degree, smoothness assumptions, and boundary orientation.
know that a connection is needed to compare fibers, vectors, or phases at different points, and that curvature measures local path dependence.
read parallel transport as the equation $\nabla_{\dot\gamma}s=0$ along a path.
recognize holonomy as closed-loop parallel transport, with Berry and Aharonov–Bohm phases as examples.
distinguish a base space, fiber, total space, and section in a first fiber-bundle example.
distinguish an isolated global phase from local $U(1)$ gauge freedom over a parameter space.
read the Berry connection as a local one-form with a gauge transformation and curvature.
identify when a loop has a winding number because the allowed space has a hole or phase circle.
recognize a first Chern number as a normalized curvature integral over a closed oriented surface.
distinguish a topological invariant from merely local, coordinate-invariant, or gauge-invariant data.

Repair links:

Numerical Readiness

You are ready for introductory computational pages if you can:

diagonalize a small Hermitian matrix by hand or with software;
check that numerical eigenvectors are normalized and mutually orthogonal;
compare a numerical answer with a limiting case or exactly solvable model;
recognize roundoff error and catastrophic cancellation in a simple calculation;
distinguish a sensitive problem from an unstable algorithm at a basic level;
distinguish discretization error from roundoff error at a basic level;
explain why a grid wavefunction needs quadrature weights in its norm;
choose an appropriate quadrature rule for a normalization or expectation-value integral;
write the standard centered second-difference approximation to $d^2/dx^2$ ;
distinguish local finite-difference stencils from global spectral expansions;
estimate when a Hamiltonian matrix should be stored sparsely rather than densely;
explain why a sparse eigensolver needs residual checks, not only stabilized eigenvalue estimates;
distinguish a norm-preserving time step from a merely stable time step;
decide whether a computation needs the full matrix exponential or only its action on a state;
map FFT array indices to positive, negative, and Nyquist momentum modes;
set up a shooting residual for a one-dimensional bound-state ODE;
distinguish spatial discretization error from time-step error in a PDE solve;
separate truncation, roundoff, solver, and statistical error sources in a reported result;
estimate observed convergence order from three refinement levels;
choose a benchmark problem that exercises the same failure mode as the target calculation;
report enough conventions that another reader can reproduce the calculation.

Repair links:

How to Interpret the Checklist

If one item in a section is unfamiliar, read the repair page and continue. If several items in the same section are unfamiliar, spend time with that section before using it inside a quantum calculation. If a page depends on a convention, read the convention page first; convention errors are often harder to spot than algebra errors.

For a volume-specific route, use the crosswalk pages rather than trying to master every mathematical topic at once.

Cross-Links

References

M. L. Boas, Mathematical Methods in the Physical Sciences, 3rd ed., Wiley, 2005.
G. Strang, Linear Algebra and Its Applications, 4th ed., Brooks/Cole, 2006.
G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed., Academic Press, 2013.
B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.

Exercises

\lvert\psi\rangle = \frac{1}{\sqrt3} \begin{pmatrix} 1\\ 1+i \end{pmatrix}.

Is $\lvert\psi\rangle$ normalized?

Solution

Compute the norm:

\lVert\psi\rVert^2 = \frac{1}{3} \left( \lvert1\rvert^2+\lvert1+i\rvert^2 \right) = \frac{1}{3}(1+2) = 1.

The vector is normalized.

In a two-dimensional Hilbert space, let

P= \begin{pmatrix} 1 & 0\\ 0 & 0 \end{pmatrix}.

Which checklist section does the test $P^2=P=P^\dagger$ belong to, and what does it show?

Solution

It belongs to linear algebra readiness. The identities show that $P$ is an orthogonal projector onto the first basis vector.

A wavefunction has $\int_{-\infty}^{\infty}\lvert\psi(x)\rvert^2\,dx=4$ . What is the simplest normalized wavefunction with the same shape?

Solution

Use

\psi_{\mathrm{norm}}(x) = \frac{1}{2}\psi(x).

Then

\int_{-\infty}^{\infty} \lvert\psi_{\mathrm{norm}}(x)\rvert^2\,dx = \frac{1}{4} \int_{-\infty}^{\infty}\lvert\psi(x)\rvert^2\,dx = 1.