Quantum Mechanics (QM1–QM17)

17 operators covering the full quantum mechanical spectrum — from the Schrödinger equation to quantum field theory. Every computation includes KO42 metric tensioner overlay.

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All QM operators are available in the free tier (10 computations/day). Results include ZeqProof cryptographic receipts.

QM1 — Schrödinger Equation

iħ ∂ψ/∂t = −ħ²/2m ∂²ψ/∂x² + Vψ

Time-dependent wave equation governing quantum state evolution. The foundation of non-relativistic quantum mechanics.

QM2 — Heisenberg Uncertainty

Δx · Δp ≥ ħ/2

Fundamental limit on simultaneous measurement precision of conjugate variables (position and momentum).

QM3 — Superposition

|ψ⟩ = ∑ c_i |φ_i⟩

Any quantum state can be expressed as a linear combination of basis states with complex coefficients.

QM4 — Entanglement

|ψ⟩ = 1/√2 (|↑⟩_A |↓⟩_B − |↓⟩_A |↑⟩_B)

Bell state — maximally entangled two-particle state. Used in ZSP key exchange (K_chaos kernel).

QM5 — Eigenvalue Equation

Ĥ|ψ⟩ = E|ψ⟩

Time-independent Schrödinger equation. Eigenvalues E are the allowed energy levels of the system.

QM6 — Antisymmetry (Pauli Exclusion)

ψ(x&sub1;, x&sub2;) = −ψ(x&sub2;, x&sub1;)

Fermionic wave functions must be antisymmetric under particle exchange.

QM7 — Spin

Ŝ²|ψ⟩ = s(s+1)ħ²|ψ⟩

Spin angular momentum eigenvalue equation. Intrinsic angular momentum quantised in half-integer or integer units.

QM8 — Quantum Tunneling

T ∝ e^{−2 ∫ √(2m(V−E))/ħ² dx}

Transmission coefficient through a potential barrier. Core component of ZSP’s K_chaos kernel for deterministic chaos generation.

QM9 — de Broglie Wavelength

λ = h/p

Matter-wave duality — every particle has an associated wavelength inversely proportional to its momentum.

QM10 — Planck-Einstein Relation

E = hν

Energy of a photon is proportional to its frequency. Foundation of quantum theory.

QM11 — Canonical Commutator

[x̂, p̂] = iħ

Position-momentum commutation relation. The algebraic foundation of quantum mechanics and the uncertainty principle.

QM12 — Dirac Equation

(iγ^μ ∂_μ − m)ψ = 0

Relativistic wave equation for spin-1/2 particles. Predicts antimatter and spin-orbit coupling.

QM13 — QED Lagrangian

ℒ = ψ̄(i∕D − m)ψ

Quantum electrodynamics Lagrangian density. Describes electron-photon interactions.

QM14 — Bose-Einstein Distribution

n_i = 1/[e^{(E_i − μ)/k_B T} − 1]

Occupation number for bosons. Governs photon statistics, superfluidity, and Bose-Einstein condensates.

QM15 — Fermi-Dirac Distribution

n_i = 1/[e^{(E_i − μ)/k_B T} + 1]

Occupation number for fermions. Governs electron distribution in metals and semiconductors.

QM16 — Heisenberg Equation of Motion

dÂ/dt = i/ħ [Ĥ, Â]

Time evolution of an observable in the Heisenberg picture.

QM17 — Born Rule

P(x) = |ψ(x)|²

Probability density is the squared modulus of the wave function. The bridge between quantum formalism and measurement.

API Usage

Call any operator via the Compute API:

curl -X POST https://zeq.dev/api/zeq/compute \
  -H "Content-Type: application/json" \
  -H "Authorization: Bearer zk_live_..." \
  -d '{"query": "Calculate electron tunneling probability through 2nm barrier", "operators": ["QM8, KO42"]}'

The 7-step wizard protocol selects the right operators automatically when you omit the operators field. KO42 is always included.