Quantum Benchmarks for Numerical Methods

Benchmark problems are controlled quantum problems used to test numerical methods before trusting them on new physics. A benchmark has a known answer, a known convergence pattern, or an independent way to detect failure.

Good benchmarks are not decorative examples. They catch wrong boundary rows, missing quadrature weights, unstable time steps, mislabeled eigenstates, incorrect units, broken symmetry, phase errors, and false convergence.

The examples below pair exact formulas with numerical checks. Follow the linked model pages for their physical derivations.

Helpful background. Review convergence tests to vary one approximation parameter while keeping the physical problem fixed.

Choosing a benchmark with a known answer

A good benchmark should have:

• a clearly specified Hamiltonian, domain, and boundary conditions;
• an exact or highly trusted reference answer;
• a known scale for energy, length, and time;
• observables that test the feature you care about;
• a refinement path with expected behavior;
• failure modes similar to the production calculation.

Do not benchmark only the easy part of the code. A time-evolution code needs phase and conservation tests, not just a static eigenvalue. A PDE solver with boundary conditions needs benchmarks that actually exercise boundaries. A Monte Carlo estimator needs known statistical behavior and bias checks.

Benchmark Ladder

Use benchmarks in layers:

Layer	Purpose	Example
algebraic identity	catch assembly mistakes	Hermiticity, trace, normalization
small finite model	compare with dense exact result	two-level system
continuum exact solution	test discretization	infinite well, oscillator
scattering exact solution	test current and matching	rectangular barrier
radial singular problem	test endpoint handling	hydrogen radial equation
convergence benchmark	test observed order	grid and time-step refinement
regression benchmark	prevent future drift	saved validated cases

A mature code or notebook usually uses several layers, not one heroic benchmark.

Infinite Square Well

The Infinite Square Well tests hard-wall boundary conditions, kinetic-energy stencils, eigenvalue ordering, normalization, and node counting.

For $0\lt x\lt L$ with $\psi(0)=\psi(L)=0$,

$$E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}, \qquad n=1,2,3,\ldots$$

and

$$\psi_n(x) = \sqrt{\frac{2}{L}} \sin\frac{n\pi x}{L}.$$

Numerical checks:

• low-lying eigenvalues converge under grid refinement;
• the $n$th state has $n-1$ interior nodes;
• eigenvectors are orthonormal in the discrete inner product;
• boundary values satisfy the hard-wall condition;
• high-energy states near the grid cutoff are not overinterpreted.

For the three-point finite-difference Hamiltonian, compare separately with the exact discrete sine spectrum in Finite Difference Methods. Agreement with that spectrum checks assembly and the eigensolver. Its difference from $E_n$ measures spatial discretization error. These two comparisons diagnose different failures.

This benchmark is excellent for detecting boundary-row errors. It is less useful for testing smooth potentials because the potential is zero inside and infinite at the walls.

Harmonic Oscillator

The Quantum Harmonic Oscillator tests smooth confinement, basis truncation, grid-domain truncation, Gaussian tails, ladder structure, and long-time phase evolution.

The Hamiltonian is

$$H = \frac{p^2}{2m} + \frac12m\omega^2x^2.$$

The exact spectrum is

$$E_n = \hbar\omega \left( n+\frac12 \right), \qquad n=0,1,2,\ldots$$

The oscillator length is

$$\ell = \sqrt{\frac{\hbar}{m\omega}}.$$

Numerical checks:

• energy spacing is constant and equal to $\hbar\omega$;
• the ground-state energy is $\hbar\omega/2$;
• wavefunction tails are negligible at the finite-box boundary;
• parity alternates between even and odd states;
• expectation values satisfy the virial relation $\langle T\rangle=\langle V\rangle=E_n/2$;
• a superposition revives with the correct relative phases.

This benchmark is sensitive to both grid spacing and finite-domain error. A box that is too small can spoil the Gaussian tail even when the grid is fine.

Rectangular Barrier

Rectangular Barrier Tunneling tests matching conditions, current normalization, evanescent waves, transfer matrices, and exponentially small transmission.

For

$$V(x)= \begin{cases} 0, & x\lt0,\\ V_0, & 0\lt x\lt a,\\ 0, & x\gt a, \end{cases}$$

with $0\lt E\lt V_0$, define

$$\kappa = \frac{\sqrt{2m(V_0-E)}}{\hbar}.$$

The exact transmission coefficient for equal potentials on the two sides is

$$T = \left[ 1+ \frac{V_0^2\sinh^2(\kappa a)} {4E(V_0-E)} \right]^{-1}.$$

Numerical checks:

• transmission and reflection satisfy $T+R=1$ for a conservative calculation;
• current ratios, not raw amplitudes, define probabilities;
• the opaque-barrier limit gives $T\propto e^{-2\kappa a}$;
• matching at discontinuities is implemented consistently;
• tiny transmissions are larger than roundoff and truncation uncertainty.

This benchmark is excellent for detecting incorrect flux normalization and unstable treatment of growing and decaying exponentials.

Hydrogen Radial Equation

The Hydrogen Atom tests radial coordinates, singular endpoints, angular-momentum barriers, reduced mass, and Coulomb tails.

After separation, the reduced radial function $u(r)=rR(r)$ satisfies

$$\left[ - \frac{\hbar^2}{2\mu} \frac{d^2}{dr^2} + \frac{\hbar^2\ell(\ell+1)}{2\mu r^2} - \frac{e^2}{4\pi\epsilon_0 r} \right]u(r) = Eu(r).$$

The bound-state energies are

$$E_n = - \frac{\mu e^4}{2(4\pi\epsilon_0)^2\hbar^2} \frac{1}{n^2}, \qquad n=1,2,\ldots$$

Numerical checks:

• regularity at $r=0$ is enforced;
• the radial measure is handled consistently;
• $u(0)=0$ and the tail decays at large $r$;
• degeneracy across allowed $\ell$ values is reproduced in the pure Coulomb model;
• reduced mass is used when comparing physical hydrogen;
• highly excited states require larger radial domains.

This benchmark is more demanding than the infinite well or oscillator because the coordinate singularity and long tail both matter.

Two-Level System

The Two-Level System tests finite-dimensional linear algebra, exact matrix exponentials, unitarity, phase conventions, and time stepping without spatial discretization.

With real $c_0$ and $\mathbf b$, a general Hermitian two-level Hamiltonian can be written

$$H = c_0I + \mathbf b\cdot\boldsymbol\sigma.$$

The eigenvalues are

$$E_\pm = c_0\pm\lVert\mathbf b\rVert.$$

For the special Hamiltonian

$$H = \frac{\hbar\Omega}{2}\sigma_x,$$

starting in one computational basis state gives oscillatory population transfer:

$$P_{\mathrm{flip}}(t) = \sin^2\frac{\Omega t}{2}.$$

Numerical checks:

• eigenvalues match $c_0\pm\lVert\mathbf b\rVert$;
• time evolution remains unitary;
• Rabi oscillation frequency is correct;
• global phase from $c_0I$ does not affect probabilities;
• time-step error appears as phase error before norm error in some methods.

This benchmark is ideal for separating time-integration errors from spatial discretization errors because there is no spatial grid.

A conserved norm with the wrong transition probability

The energy offset supplies an additional test. Take $H_c=cI+\hbar\Omega\sigma_x/2$, with constant real $c$ and $\Omega>0$, and prepare $|0\rangle$. Exact evolution still gives $P_{\mathrm{flip}}(t)=\sin^2(\Omega t/2)$: the common energy shift contributes only a global phase. An approximate step need not preserve this independence.

Use the eigenmode multiplier derived in Crank–Nicolson Method. After $n$ equal steps, the two eigenphases give

$$\theta_\pm=2\arctan\!\left[\frac{(c\pm\hbar\Omega/2)\Delta t}{2\hbar}\right],$$ $$P_{\mathrm{flip}}^{\mathrm{CN}}(n\Delta t) =\sin^2\!\left[\frac n2(\theta_+-\theta_-)\right].$$

The arctangent is nonlinear, so a common shift can change the numerical relative phase. In units with $\hbar=\Omega=1$, choose $\Delta t=0.2$ and $n=16$, so $t=3.2$. The exact probability is approximately $0.999147$ for either offset:

Propagation	Flip probability
Direct Crank–Nicolson, $c=0$	$0.999223$
Direct Crank–Nicolson, $c=10$	$0.514933$
Crank–Nicolson after removing $10I$, with its phase restored	$0.999223$

Every row preserves the state norm and the corresponding constant-Hamiltonian energy expectation in exact arithmetic. Those invariants therefore miss the large probability error in the second row.

To obtain the last row, write $\psi(t)=e^{-ict/\hbar}\chi(t)$ and propagate $\chi$ with $H_c-cI$. Restore the scalar phase when an amplitude comparison needs it; see Matrix Exponentials. This removes the artificial dependence on the known common offset, but the centered step still has finite-step error. Refine $\Delta t$ at fixed final time and compare the transition probability, rather than accepting the conserved norm as convergence.

Free Particle on a Periodic Grid

A periodic free particle tests FFT conventions, momentum indexing, dispersion, and periodic wraparound.

On a periodic interval of length $L$, Fourier modes have wavenumbers

$$k_n = \frac{2\pi n}{L}.$$

The kinetic energies are

$$E_n = \frac{\hbar^2k_n^2}{2m}.$$

Numerical checks:

• FFT frequency ordering matches the kinetic-energy array;
• positive and negative modes with the same $\lvert n\rvert$ are degenerate;
• free wave packets disperse at the correct rate;
• periodic wraparound is recognized as a numerical boundary effect;
• Parseval normalization agrees between position and momentum grids.

This benchmark is especially useful before using split-operator evolution.

Benchmark Selection

Choose a benchmark that stresses the same feature as the intended calculation.

Production feature	Benchmark to start with
hard-wall boundary eigenproblem	infinite square well
smooth bound states	harmonic oscillator
Fourier-grid dynamics	free particle on periodic grid
tunneling or evanescent tails	rectangular barrier
radial coordinates	hydrogen radial equation
finite-dimensional time stepping	two-level system
sparse eigensolver	finite-difference well or oscillator
quadrature-weighted basis	oscillator or hydrogen matrix elements

Passing an easy benchmark is useful but not sufficient. The benchmark should be close enough to expose the same numerical risks.

Choosing an accuracy target

Before running a benchmark, decide what “passes” means. Examples:

• first five oscillator energies agree within the estimated grid error;
• norm drift stays below $10^{-10}$ over a specified time interval;
• observed convergence order is within a stated tolerance;
• transmission satisfies $T+R=1$ to the expected accuracy;
• the hydrogen $n=2$ degeneracy is resolved only at the controlled error scale;
• independent methods agree within their combined error estimates.

State the target before comparing results, then refine the calculation to check that agreement improves as expected.

Common Mistakes

• Using a benchmark that does not test the feature that matters in production.
• Comparing to formulas in different units or conventions.
• Checking only the ground-state energy and ignoring wavefunctions or observables.
• Trusting high-energy grid eigenstates near the cutoff.
• Ignoring finite-box effects in oscillator and hydrogen benchmarks.
• Testing time evolution only at short times when phase error has not accumulated.
• Treating a benchmark plot as a substitute for a convergence table.
• Updating reference benchmark values after code changes without investigating the change.

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, 2007.
• T. Pang, An Introduction to Computational Physics, 2nd ed., Cambridge University Press, 2006.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• D. J. Tannor, Introduction to Quantum Mechanics: A Time-Dependent Perspective, University Science Books, 2007.
• J. M. Thijssen, Computational Physics, 2nd ed., Cambridge University Press, 2007.

Exercises

1. Which benchmark best tests hard-wall boundary rows in a finite-difference Hamiltonian?

Solution

The infinite square well is the natural first benchmark. Its energy quantization comes entirely from Dirichlet boundary conditions, and the exact eigenfunctions vanish at both endpoints. Boundary-row mistakes show up clearly in the spectrum and eigenfunctions.

2. A harmonic-oscillator grid calculation gives accurate low energies but changes when the box is enlarged. What error source is indicated?

Solution

The calculation is likely limited by finite-domain error. The oscillator wavefunction tails are still feeling the boundary. Increase the box size until low-energy results are insensitive to boundary placement, then refine the grid spacing.

3. Why is a two-level system a useful benchmark for time-stepping methods? What additional error can be exposed by adding a common energy offset?

Solution

It has exact finite-dimensional dynamics and no spatial discretization. Therefore any error in norm, phase, transition probability, or energy expectation comes from the time integrator or matrix exponential implementation, not from grid spacing or boundary conditions. A common offset must leave exact transition probabilities unchanged. If the computed probabilities change, the numerical step has introduced an energy-origin dependence; the Crank–Nicolson example above shows that norm and energy conservation need not detect it. Removing the offset does not eliminate the remaining time-step error.

4. In a rectangular barrier benchmark with equal potentials on both sides, why should $T+R=1$?

Solution

The Hamiltonian is conservative and Hermitian, and there is no absorption. Probability current is conserved. With equal asymptotic wave numbers, the incident current is split into reflected and transmitted currents, so the corresponding current ratios satisfy $T+R=1$.

5. Why is hydrogen a harder benchmark than the infinite square well?

Solution

Hydrogen has a singular Coulomb potential at the origin, an angular-momentum barrier in the radial equation, a long tail, and radial measure issues. The infinite square well has simple hard-wall boundaries and a constant interior potential. Hydrogen therefore tests endpoint regularity, domain truncation, coordinate factors, and degeneracy more severely.