Exact references

Question: does the code match a result known in closed form, or computable exactly?

physverify ships a collection of exact results that people kept writing by hand. Each one is tested in physverify's own test suite against an independent computation (finite differences, enumeration, orbit integration).

physverify references

lists them; help(physverify.references.<name>) gives the formula and its source.

Example: a Schroedinger solver against the Morse levels

"""Exact references: compare a finite-difference Schroedinger solver with the Morse levels."""
import numpy as np

from physverify import references as R

D, a = 12.0, 1.0  # hbar = m = 1
x = np.linspace(-2.5, 14.0, 3000)
h = x[1] - x[0]
V = D * (1 - np.exp(-a * x)) ** 2
off = np.full(len(x) - 1, 0.5 / h**2)
H = np.diag(1 / h**2 + V) - np.diag(off, 1) - np.diag(off, -1)
E = np.linalg.eigvalsh(H)[:4]
exact = R.morse_levels(np.arange(4), a * np.sqrt(2 * D), a * a / 2)
for v in range(4):
    print(f"v={v}: solver {E[v]:.5f}  exact {exact[v]:.5f}  difference {abs(E[v] - exact[v]):.1e}")

Output:

v=0: solver 2.32447  exact 2.32449  difference 2.0e-05
v=1: solver 6.22340  exact 6.22347  difference 6.4e-05
v=2: solver 9.12235  exact 9.12245  difference 9.7e-05
v=3: solver 11.02134  exact 11.02143  difference 8.6e-05

The differences are small and shrink as the grid is refined, as they should for a second-order finite-difference Hamiltonian.

What is available

Area	References
Statistical physics	2D Ising model by exact enumeration (up to 4x4), classical Heisenberg chain, open transverse-field Ising chain ground energy
Heat and transport	heat-equation sine modes, a moving and spreading Gaussian, the semi-infinite solid with a temperature step
Mechanics	double-pendulum accelerations and energy, cantilever tip deflection and slope
Quantum mechanics	infinite square well, square-barrier transmission, Morse levels and turning points
Gravitation	Schwarzschild quasinormal frequencies, Kerr innermost stable orbit, exact light deflection
Reactor physics	asymptotic period from the inhour equation, for either sign of reactivity

A real code used the positive-reactivity formula for a negative reactivity and was off by a factor of about 7000; the reference handles both signs.