Analytic Benchmarks

Analytic benchmarks are exact or near-exact results used to validate numerical calculations. They are valuable because they test both physics and code: boundary conditions, normalization, units, basis truncation, time evolution, and observable definitions.

The canonical derivations remain on the teaching or formula pages. This page identifies which analytic targets are useful for computational validation.

Infinite Square Well

Use for: hard-wall boundary conditions, kinetic-energy stencils, eigenvalue ordering, node counting, and grid normalization.

Target:

E_n=\frac{n^2\pi^2\hbar^2}{2mL^2}, \qquad \psi_n(x)=\sqrt{\frac{2}{L}}\sin\frac{n\pi x}{L}.

Required checks:

eigenvalues converge under grid refinement,
eigenvectors are orthonormal with the discrete inner product,
boundary values satisfy the hard-wall condition,
high-energy grid states are not overinterpreted.

Canonical link: Infinite Square Well.

Harmonic Oscillator

Use for: smooth confinement, basis truncation, parity, ladder-operator checks, and long-time phase evolution.

Target:

E_n=\hbar\omega\left(n+\frac12\right).

Required checks:

low-energy spacing approaches $\hbar\omega$ ,
eigenstates have alternating parity,
boundary-tail error is controlled,
expectation values satisfy the virial relation in stationary states.

Canonical link: Quantum Harmonic Oscillator.

Two-Level System

Use for: finite-dimensional time evolution, matrix exponentials, unitarity, global phase, and circuit-state conventions.

Target:

H=c_0I+\mathbf b\cdot\boldsymbol\sigma, \qquad E_\pm=c_0\pm\lVert\mathbf b\rVert.

For $H=\hbar\Omega\sigma_x/2$ ,

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

Required checks:

eigenvalues match the closed form,
state norm is conserved,
transition probability has the correct frequency,
global phase does not affect probabilities.

Canonical link: Two-Level System.

Free Particle and FFT Grids

Use for: Fourier conventions, momentum indexing, dispersion, and split-operator methods.

Target on a periodic interval of length $L$ :

k_n=\frac{2\pi n}{L}, \qquad E_n=\frac{\hbar^2k_n^2}{2m}.

Required checks:

FFT frequency ordering matches the kinetic-energy array,
positive and negative momentum modes have the expected degeneracy,
Parseval normalization is consistent,
packet spreading matches the analytic result before boundary wraparound.

Canonical link: Free Particle.

Rectangular Barrier

Use for: scattering matching, current conservation, evanescent waves, and tunneling exponentials.

For equal asymptotic potentials and $0<E<V_0$ ,

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

Required checks:

$R+T=1$ for a conservative calculation,
probabilities are current ratios,
opaque-barrier behavior follows the expected exponential scale,
wave-packet results agree with stationary scattering only in the correct limit.

Canonical link: Rectangular Barrier Tunneling.

Hydrogenic Bound States

Use for: radial equations, singular endpoints, Coulomb tails, degeneracy, and quadrature measures.

Target:

E_n= - \frac{\mu e^4}{2(4\pi\epsilon_0)^2\hbar^2} \frac{1}{n^2}.

Required checks:

radial normalization uses the correct measure,
endpoint regularity is enforced,
bound-state energies scale as $-1/n^2$ ,
pure Coulomb degeneracies appear only when the model assumptions are unchanged.

Canonical link: Hydrogen Atom.

Cross-Links

References

D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
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.
D. J. Tannor, Introduction to Quantum Mechanics: A Time-Dependent Perspective, University Science Books, 2007.