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|ϕ_i⟩	Superposition
QM4	|ψ⟩ = 1/√2 (|↑⟩_A|↓⟩_B − |↓⟩_A|↑⟩_B)	Bell-state entanglement
QM5	Ĥ|ψ⟩ = E|ψ⟩	Energy eigenvalue
QM6	ψ(x₁,x₂) = −ψ(x₂,x₁)	Fermionic antisymmetry
QM7	Ŝ²|ψ⟩ = s(s+1)ℏ²|ψ⟩	Spin eigenvalue
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)|²	Born probability

Runnable worked example — particle-in-a-box eigenvalue, sealed to HulyaPulse

A synchronized computation: KO42 (1.287 Hz metric tensioner) composed with QM5 (Ĥ|ψ⟩ = E|ψ⟩) for the ground-state energy of an electron in a 1 Å box. Get a free key first (POST /api/demo-key/mint), then:

curl -X POST https://zeqsdk.com/api/zeq/compute \
  -H "Authorization: Bearer zeq_ak_..." \
  -H "Content-Type: application/json" \
  -d '{
    "operators": ["KO42", "QM5"],
    "domain": "Quantum Mechanics",
    "inputs": { "n": 1, "L": 1e-10, "m": 9.109e-31 }
  }'

Real response (abridged):

{
  "value": 6.0249212e-18,
  "unit": "J",
  "operator_id": "QM5",
  "zeqState": { "operators": ["KO42", "QM5"], "domain": "Quantum Mechanics" },
  "verification": { "precision_bound": "≤ 0.1%" },
  "zeqProof": "…"
}

value is E₁ = n²h²/8mL² = 6.0249212e-18 J for the ground state — verified live against the closed-form eigenvalue.

Solver coverage

The Quantum Mechanics domain has a live closed-form solver — the particle-in-a-box example above returns a real number today. Not every catalogued operator is wired to it: operators without solver support return no-match from /api/zeq/compute (unit no-match, no value). The catalogue rows are exact and documented regardless.

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.