Model Systems, Inside Quantum Mechanics

Nuclei Obey Quantum Mechanics, So We Solve It

Eigenstates, tunnelling and wavepacket dynamics on grids, each checked against an exact or published answer.

A researcher at a desk with a display showing a teal quantum wavepacket tunnelling through a potential barrier

Five Tools, Each with a Closed-Form Check

Live

Grid

Dense diagonalisation in 1D and Lanczos in 2D and 3D, with a discrete variable representation (DVR) or 3- to 9-point finite-difference stencils.

Model

Box, harmonic, Morse, double well, Coulomb and Eckart systems, each compared with the exact answer.

Tunnel

Eckart closed form and a transfer matrix, with Wigner and Eckart tunnelling corrections.

Wavepacket

Split-operator propagation, autocorrelation, spectrum and peak positions, with CPU or CUDA FFT.

Anharmonic

Vibrational levels, zero-point energy and the anharmonicity constants ωe and ωexe from a bond scan.

Verified Against Exact Answers

The solvers are tested on problems whose answers are known in closed form.

Our own FFT (radix-2, mixed radix and Bluestein) matches a direct transform to 1.75e-15
Threaded, with a runtime AVX2 stage and a Stockham CUDA kernel of 0.38 ms on the RTX 3050

Relative errors as recorded in the engine self-test.
Check	Error	Reference
Harmonic Levels, v = 0 to 9	8.2e-14	Exact, E = ħω(v + ½)
Morse Levels, v = 0 to 9	4.3e-10	Morse 1929
Hydrogen 1s to 3d Radial	1.66e-13	Exact, E = −Z²/2n²
Eckart Transmission	1.4e-10	Eckart 1930, transfer matrix
Asymmetric Eckart	7.4e-11	Johnston and Heicklen 1962

A Result That Shows Its Own Gap

Hydrogen from an HF/STO-3G bond scan gives ωe = 5453.9 cm⁻¹ against 4401.21 cm⁻¹ measured, +23.9 percent. The solver is right for the potential it was given. The potential is the weak link, and the self-test prints the difference as information so nobody mistakes it for a pass.

Two scientists at a meeting table comparing printed result charts and a laptop showing two nearly identical curves

Model Systems Questions

What are Model Systems?

Nuclear quantum mechanics on grids: the Schrödinger equation for nuclei moving on a potential, solved numerically. It covers eigenstates, tunnelling through barriers and the time evolution of wavepackets. It is not electronic structure.

Can it use a potential from a quantum chemistry scan?

Yes. The anharmonic tool takes an HF/STO-3G bond scan, or points you supply, and returns vibrational levels, zero-point energy, ωe and ωexe.

Is the H2 result accurate?

It is faithful to the potential it is given. Using the HF/STO-3G scan, ωe is 5453.9 cm⁻¹ against 4401.21 cm⁻¹ experimental, +23.9 percent. That gap belongs to the Hartree-Fock potential, and the self-test prints it as information, not as a pass.

Does it run on the GPU?

The wavepacket solver has its own CUDA FFT, but we have not measured it as faster than the CPU, so we do not claim a speed-up.

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