Benchmark Problems

Benchmark problems are controlled calculations used to decide whether a notebook, solver, or plot can be trusted. The Mathematical Toolkit benchmark page owns the general numerical-method guidance. This page specializes that guidance to the Wave Mechanics and Model Systems volume.

A benchmark should state the Hamiltonian, boundary conditions, numerical method, reference result, refinement variable, and pass criterion before the computation is run.

Benchmark Suite

ID	Benchmark	Numerical Task	Reference Result	Pass Criterion	Canonical Page
WMCS-B001	Infinite square well	finite-difference Hamiltonian eigenproblem	$E_n=n^2\pi^2\hbar^2/(2mL^2)$	low-lying energies converge under grid refinement and eigenvectors are orthonormal in the grid inner product	Infinite Square Well
WMCS-B002	Harmonic oscillator	diagonalize a grid or basis Hamiltonian	$E_n=\hbar\omega(n+1/2)$	low levels show constant spacing, expected parity, and negligible boundary-tail error	Quantum Harmonic Oscillator
WMCS-B003	Finite square well	find bound states by shooting or diagonalization	matching equations and bound-state count	energies satisfy interface conditions and the number of bound states changes correctly with well depth	Finite Square Well
WMCS-B004	Free Gaussian packet	evolve a localized wave packet	analytic width and conserved norm	norm drift stays below the stated tolerance and $\sigma_x(t)$ follows the analytic spreading law	Wave Packet Spreading
WMCS-B005	Rectangular barrier	propagate or match a scattering state	current conservation and stationary transmission coefficient	$R+T=1$ within error and narrow packets agree with stationary transmission after momentum averaging	Rectangular Barrier Tunneling
WMCS-B006	Landau levels	compute magnetic-field spectrum or construct Landau-gauge states	equal spacing $\hbar\omega_c$ and degeneracy $A/(2\pi\ell_B^2)$	spacing, gauge convention, and finite-area degeneracy scaling are reported and checked	Landau Levels

Acceptance Criteria

Every benchmark run should include:

• the physical parameters and unit convention;
• the grid, basis, time step, solver tolerance, and finite domain;
• the observable being compared, not only a plot;
• an error estimate or convergence table;
• a statement of which effects are finite-size, discretization, solver, or roundoff errors.

Typical pass criteria are scale dependent. A useful benchmark might require low-lying eigenvalue errors below $10^{-4}$ in chosen dimensionless units, norm drift below $10^{-10}$ over a stated time window, or observed second-order convergence for a second-order finite-difference stencil. The numerical tolerance should be justified by the purpose of the notebook.

Reference Checks

For finite-difference eigenproblems, check both energies and vector normalization. For a grid with spacing $\Delta x$, a one-dimensional normalized state should satisfy

$$\sum_j \lvert\psi_j\rvert^2\Delta x \approx 1.$$

For closed-system time evolution, norm conservation is necessary but not sufficient. Also monitor energy expectation for time-independent Hamiltonians:

$$\langle H\rangle(t) =\langle\psi(t)\vert H\vert\psi(t)\rangle.$$

For scattering benchmarks, use probability current. If the asymptotic wave numbers are $k_{\mathrm{in}}$ and $k_{\mathrm{out}}$, amplitudes alone do not define transmission:

$$T =\frac{j_{\mathrm{out}}}{j_{\mathrm{in}}}.$$

For Landau-level calculations in a bulk region of area $A$,

$$N_\phi =\frac{A}{2\pi\ell_B^2}, \qquad \ell_B=\sqrt{\frac{\hbar}{\lvert q\rvert B}}.$$

Finite geometry, edges, and boundary conditions can shift finite-size counts. A benchmark should state whether it tests the bulk formula, a finite rectangle, or a specific boundary convention.

Benchmark Reports

A short benchmark report should include a table like this:

Run	Refinement variable	Observable	Reference	Error	Interpretation
example	grid spacing $h$	$E_1$	exact well energy	relative error	checks kinetic stencil and boundary rows
example	time step $\Delta t$	norm drift	zero drift	maximum drift	checks time-stepper unitarity
example	domain size $L_{\mathrm{box}}$	oscillator $E_0$	$\hbar\omega/2$	absolute error	checks tail truncation

The interpretation column matters. A benchmark is not only a number; it is a diagnosis of what the computation has or has not tested.

Common Mistakes

• Passing a benchmark at one grid size and calling it convergence.
• Checking energy but not eigenfunction normalization or orthogonality.
• Reporting norm conservation while the wave-packet width, phase, or transmission is wrong.
• Comparing a finite-box calculation with an infinite-line formula without a domain-size study.
• Forgetting that scattering probabilities are current ratios.
• Testing a gauge-dependent wavefunction without stating the gauge.
• Moving benchmark tolerances after a code change to make the result pass.

Where This Is Used

• Numerical Notebooks Index lists the notebook artifacts that should use these benchmarks.
• Problem-Solving Patterns explains the analytic and numerical checks that benchmarks formalize.
• Visualization Gallery records which checked outputs have been promoted to reference figures.
• Exercise Sets includes computational exercises that can be promoted to benchmark notebooks.
• Spectra and Eigenfunctions Table supplies exact formulas for benchmark comparisons.
• Convergence Tests explains how to vary grid spacing, domain size, basis size, time step, and solver tolerance.
• Error Estimates explains how to turn refinement behavior into uncertainty statements.

References

• J. M. Thijssen, Computational Physics, 2nd ed., Cambridge University Press, 2007.
• R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, 2007.
• D. J. Tannor, Introduction to Quantum Mechanics: A Time-Dependent Perspective, University Science Books, 2007.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.

Exercises

1. A finite-difference infinite-well notebook reports the first five energies but does not check eigenvector orthogonality. Why is the benchmark incomplete?

Solution

The spectrum alone can look plausible even if the discrete inner product, boundary rows, or normalization convention is wrong. Orthonormality checks whether the finite matrix and grid weights correctly represent the Hilbert-space inner product. For a Hermitian discretized Hamiltonian, low-lying eigenvectors should be mutually orthonormal in the correct discrete inner product.

2. A barrier-scattering notebook finds $R+T=0.97$ for a conservative rectangular barrier. List two possible diagnoses.

Solution

The packet may not have fully separated from the barrier or boundaries, so reflected and transmitted probabilities have not been measured over complete regions. Alternatively, the numerical time evolution may not conserve norm well enough, absorbing boundaries may be present, or the coefficients may have been computed from amplitudes instead of currents.