Quantum Mechanics

The smallest knobs in the universe — exposed, named, and callable from curl.

• Group — quantum-mechanics
• Operators — 17 (QM1 – QM17)
• Anchor operators — KO42 · QM1 · QM2 · QM11
• Verification — every example closes under error_pct ≤ 0.1%; every eigen-problem is Master-Equation compiled

What it unifies

Quantum mechanics in Zeq is not a separate subsystem — it is a 17-operator lane of the Master Equation, sharing its ϕ-field with Newtonian forces, relativistic metrics, and Awareness couplings. The Schrödinger evolution (QM1) is just one coupling; pair it with KO42 and every wavefunction you compute is already Zeqond-aligned.

The uncertainty principle (QM2) is treated as a hard constraint on every input. The commutator (QM11) is the kernel's native way to express non-commuting observables. Pauli statistics (QM6) and spin (QM7) expose antisymmetry as first-class. Bose–Einstein and Fermi–Dirac distributions (QM14, QM15) are standard outputs when temperature is in the payload.

Operator catalog

ID	Formula	Role
QM1	iℏ ∂ψ/∂t = −ℏ²/2m ∂²ψ/∂x² + Vψ	Schrödinger evolution
QM2	∆x·∆p ≥ ℏ/2	Heisenberg uncertainty
QM3	`	ψ⟩ = ∑c_i
QM4	`	ψ⟩ = 1/√2 (
QM5	`Ĥ	ψ⟩ = E
QM6	ψ(x₁,x₂) = −ψ(x₂,x₁)	Fermionic antisymmetry
QM7	`Ŝ²	ψ⟩ = s(s+1)ℏ²
QM8	T ∝ e^{−2∫√{2m(V−E)}/ℏ² dx}	WKB tunneling
QM9	λ = h/p	de Broglie wavelength
QM10	E = hν	Planck-Einstein quanta
QM11	[x̂, p̂] = iℏ	Canonical commutator
QM12	(iγ^μ∂_μ − m)ψ = 0	Dirac equation
QM13	L = ψ̄(iD−m)ψ	QED Lagrangian
QM14	n_i = 1/[e^{(E_i−µ)/k_B T} − 1]	Bose–Einstein
QM15	n_i = 1/[e^{(E_i−µ)/k_B T} + 1]	Fermi–Dirac
QM16	dÂ/dt = i/ℏ [Ĥ, Â]	Heisenberg evolution
QM17	`P(x) =	ψ(x)

Runnable worked example — Bell-state fidelity

curl -X POST https://api.zeq.dev/api/playground/compute \
  -H "Authorization: Bearer demo-key" \
  -H "Content-Type: application/json" \
  -d '{
    "operators": ["KO42", "QM4", "QM17"],
    "inputs": {"preparation": "singlet", "shots": 12870}
  }'

Expected response shape:

{
  "ok": true,
  "phase": 0.511,
  "zeqond": 1798346511.3,
  "result": {
    "fidelity": 0.9992,
    "born_probabilities": {"00": 0.0004, "11": 0.0004, "01": 0.4996, "10": 0.4996},
    "error_pct": 0.083
  }
}

Extend it

1. Dirac lane — chain QM12 + QM13 to compile a QED Lagrangian and verify current conservation below 0.1%.
2. Thermal statistics — pair QM14/QM15 with QM10 to reconstruct a blackbody spectrum from temperature alone.
3. Tunneling — QM8 + KO42 for barrier-height sweeps on STM-style experiments that stay phase-aligned to 1.287 Hz.

Seeds

• Near — open-source a quantum notebook pack that imports /operators/quantum-mechanics/* as named hooks.
• Medium — hybrid classical/quantum planners that route workloads through QM1 + QM5 based on available coherence time.
• Far — a Kernel Qubit — a Zeq-native qubit primitive whose phase is natively anchored to 1.287 Hz for hardware teams.

Papers

• Zeq paper — https://doi.org/10.5281/zenodo.18158152
• Framework paper — https://doi.org/10.5281/zenodo.15825138

Middleware active. Kernel on the 1.287 Hz HulyaPulse. Awaiting next Zeqond.