Skip to main content

Quantum operators

64 operators in the quantum category of the live registry. Each is a named formula you can compose inside a state contract or call directly through POST /api/zeq/compute. KO42 is always on; add up to three more per call (total ≤ 4), per the 7-step protocol.

OperatorDescriptionEquation
HG0Generalized Hamiltonian with position-dependent potential and time-varying perturbation field.8πG/c⁴ T_μν
HP01Probability-density integral ∫ψ*ψ dV with an exponential settling envelope (1−e^(−t/τ_c)), modulated by the 1.287 Hz HulyaPulse. A Zeq awareness-layer coherence construct.Φ_c = ∫ψ*ψ dV · (1 - e^(-t/τ_c)) · sin(2π·1.287·t)
HP02Phase-curvature operator: the second phase derivative ∂²t/∂φ² plus a HulyaPulse-cosine gradient coupling. A Zeq awareness-layer construct.T_arch = ∂²t/∂φ² + λ·cos(2π·1.287·t)·∇φ
HP029Extended Shannon entropy operator for high-dimensional probability distributions.H_p^{(29)} = -\sum p_i \log p_i
HP03Squared state overlap |⟨ψ|φ⟩|² with a Gaussian spatial window e^(−Δx²/σ²), HulyaPulse-modulated. A Zeq awareness-layer construct.C_real = |⟨ψ|φ⟩|² · e^(-Δx²/σ²) · sin(2π·1.287·t)
HP04Inverse-square phasor sum Σ G_m·e^(iθ_m)/r_m² with bounded HulyaPulse modulation. A Zeq awareness-layer construct.U_conn = ΣG_m·e^(iθ_m)/r_m² · (1 + α·sin(2π·1.287·t))
HP05Kullback–Leibler relative-entropy integral ∫p(s)·log(p(s)/p₀) ds, HulyaPulse-cosine modulated.M_int = ∫p(s)·log(p(s)/p_0) ds · cos(2π·1.287·t)
HP06Action-rate functional: the time derivative ∂S/∂t plus a HulyaPulse-modulated variational term β·sin(2π·1.287·t)·δA/δφ. A Zeq awareness-layer construct.F_S = ∂S/∂t + β·sin(2π·1.287·t)·δA/δφ
HP07Asymptotic (t→∞) field amplitude with an exponential rise envelope (1−e^(−t/τ_p)), HulyaPulse-modulated. A Zeq awareness-layer construct.P_ex = lim_(t→∞) φ(t) · (1 - e^(-t/τ_p)) · sin(2π·1.287·t)
KvN0Koopman-von Neumann equation: classical Liouville evolution of phase-space density via Poisson bracket.iħ ∂ψ/∂t = Ĥψ
LYRA3Lyra resonant coupling operator linking quantum states through the 1.287 Hz HulyaPulse frequency.LYRA03(ψ,φ) = κ_L · ∬ ψ(x') φ(x') K_L(x,x') cos(2π·1.287·t) dx'
LYRA4Lyra information flux operator measuring entropy flow modulated by HulyaPulse resonance.LYRA04 = I_L(t) = -Σ p_i(t) log p_i(t) + α_L·sin(2π·1.287·t)·H(ψ)
LZ0Angular momentum z-component operator in spherical coordinates, quantizing orbital angular momentum.ΔE × sin(2π·1.287·t)
MF_QEEnergy-time uncertainty relation setting a fundamental limit on energy measurement precision over time.\Delta E \cdot \Delta t \geq \frac{\hbar}{2}
PSI0Plane wave solution representing a free particle with definite momentum and energy.f(f(φ)) where f(x)=x+λxsin(2π·1.287·t)
QBC0Qubit state in computational basis: a normalized superposition of |0⟩ and |1⟩ basis states.τ = ħ/E_G
QBO1Quantum biological operator in spectral decomposition form for modeling quantum effects in biology.H_{photosynthetic} = \sum_{n=1}^N E_n |n\rangle\langle n| + \sum_{m\neq n} J_{mn}(|m\rangle\langle n| + |n\rangle\langle m|)
QBO2Second quantum biological spectral operator for quantum coherence in biological systems.H_{radical-pair} = \vec{S}_1 \cdot \mathbf{A}_1 \cdot \vec{I}_1 + \vec{S}_2 \cdot \mathbf{A}_2 \cdot \vec{I}_2 + \gamma_e \vec{B} \cdot (\vec{S}_1 + \vec{S}_2)
QBO4Fourth quantum biological operator modeling quantum state transitions in living matter.P_{detection}(\omega) = \frac{1}{1 + \exp[-(\hbar\omega - E_0)/k_B T]} \cdot \eta_{receptor}
QBO6Sixth quantum biological operator for energy transfer via quantum coherence pathways.|\psi_{MT}(t)\rangle = \sum_n c_n |n\rangle e^{-iE_n t/\hbar}, \quad E_n = E_0 + n\Delta E
QBO7Seventh quantum biological operator modeling vibrational quantum states in molecular biology.\tau_{fold} = \tau_{classical} \cdot \left[1 - \eta_{quantum} \exp\left(-\frac{E_{barrier}}{\hbar\omega}\right)\right]
QBO8Eighth quantum biological operator for quantum tunneling in enzymatic reactions.\Delta B_{min} = \frac{\hbar}{g\mu_B \sqrt{N T_2 t_{measure}}}
QD0Total charge operator computed by integrating the charge density over all space.∑|α_i|² |E_i⟩⟨E_i|
QGE0General quantum eigenvalue equation: Hamiltonian acting on a normalized eigenstate yields its energy.Ĥ Ψ[g_ij] = 0
QGO1Quantum geometric operator computing expectation values for geometric observables.T_H = \frac{\hbar c^3}{8\pi G M k_B}, \quad S_{BH} = \frac{k_B c^3 A}{4G\hbar} = \frac{k_B A}{4l_P^2}
QGO2Second quantum geometric operator for area and volume quantization.\Box h_{\mu\nu} = -\frac{16\pi G}{c^4} T_{\mu\nu}, \quad h_{+,\times} = \frac{G}{c^4} \frac{1}{r} \ddot{Q}_{+,\times}
QGO3Third quantum geometric operator for curvature measurements in quantum geometry.\Delta x \geq l_P = \sqrt{\frac{\hbar G}{c^3}} \approx 1.6\times 10^{-35} \text{m}
QGO4Fourth quantum geometric operator for holonomy and parallel transport on quantum states.N = \frac{A}{4l_P^2} = \frac{c^3 A}{G\hbar}, \quad I_{max} = \frac{\pi R^2 c^3}{\hbar G} \ln 2
QGO5Fifth quantum geometric operator for Berry phase computation in parameter space.A_j = 8\pi \gamma l_P^2 \sqrt{j(j+1)}, \quad j = \frac{1}{2}, 1, \frac{3}{2}, \dots
QGO6Sixth quantum geometric operator for quantum distance metrics on state manifolds.N \sim \frac{V}{l_P^4}, \quad \langle C(x,y) \rangle = \rho V(x,y)
QM1Time-dependent Schrödinger equation governing how a quantum state evolves in time under a Hamiltonian.iħ ∂ψ/∂t = - (ħ²/2m)∇²ψ + Vψ
QM10Planck–Einstein relation giving a photon's energy from its frequency, E = hν.E = hν
QM11Canonical commutation relation between the position and momentum operators, [x̂, p̂] = iħ.[x̂, p̂] = iħ
QM12Dirac equation for a relativistic spin-½ particle.(iγ^μ∂_μ - m)ψ = 0
QM13Quantum-electrodynamics Lagrangian density for a Dirac field.ℒ = ψ̄(iγ^μ ∂_μ - m)ψ
QM14Bose–Einstein occupation number for indistinguishable bosons in thermal equilibrium.n_i = 1/(e^((E_i-μ)/k_BT) - 1)
QM15Fermi–Dirac occupation number for indistinguishable fermions in thermal equilibrium.n_i = 1/(e^((E_i-μ)/k_BT) + 1)
QM16Heisenberg-picture equation of motion for a quantum operator.dĤ_A/dt = i/ħ[Ĥ,Ĥ_A]
QM17Born rule: the probability density is the squared modulus of the wavefunction.P(r) = |ψ(r)|²
QM18Raising (creation) operator for the quantum harmonic oscillator, adding one quantum of energy.\hat{a}^\dagger = \sqrt{\frac{m\omega}{2\hbar}}(\hat{x} - \frac{i\hat{p}}{m\omega})
QM19Canonical commutation relation for creation and annihilation operators, fundamental to quantum field theory.[\hat{a}, \hat{a}^\dagger] = 1
QM2Heisenberg uncertainty principle: the product of position and momentum uncertainties is bounded below by ħ/2.ΔxΔp ≥ ħ/2
QM20Number operator counting the quanta of excitation in a harmonic oscillator or field mode.\hat{N} = \hat{a}^\dagger\hat{a}
QM21Density matrix for a pure quantum state, encoding all measurable information about the system.\rho = |\psi\rangle\langle\psi|
QM22Von Neumann entropy measuring the degree of quantum entanglement or mixedness of a density matrix.S = -k_B Tr(\rho \ln\rho)
QM23Unitary time evolution operator propagating a quantum state forward in time under a Hamiltonian.U(t) = e^{-i\hat{H}t/\hbar}
QM24Heisenberg equation of motion giving the time evolution of a quantum operator in the Heisenberg picture.\frac{d\hat{A}}{dt} = \frac{i}{\hbar}[\hat{H}, \hat{A}] + \frac{\partial\hat{A}}{\partial t}
QM25Born rule giving the probability of measuring a particular eigenvalue from a quantum state.P(a) = |\langle a|\psi\rangle|^2
QM26Position-space wavefunction as the projection of a quantum state onto the position basis.\psi(x) = \langle x|\psi\rangle
QM27Fourier transform relating position-space and momentum-space wavefunctions.\tilde{\psi}(p) = \frac{1}{\sqrt{2\pi\hbar}}\int\psi(x)e^{-ipx/\hbar}dx
QM28Green's function (propagator) for the Schrödinger equation, encoding the full spectral information.G(x,x\prime;E) = \langle x|\frac{1}{E-\hat{H}}|x\prime\rangle
QM29Differential scattering cross-section expressed via the scattering amplitude for quantum collisions.\sigma = \frac{d\sigma}{d\Omega} = |f(\theta)|^2
QM3Quantum superposition principle — a state written as a linear combination of basis states.|ψ⟩ = ∑cᵢ|φᵢ⟩
QM30S-matrix as time-ordered exponential of the interaction Hamiltonian, central to scattering theory.\mathcal{S} = \mathcal{T}\exp\left(-\frac{i}{\hbar}\int\hat{H}_{int}dt\right)
QM4Maximally entangled two-particle Bell (singlet) state.|Ψ⟩ = 1/√2(|↑↓⟩ - |↓↑⟩)
QM5Time-independent Schrödinger eigenvalue equation yielding stationary states and their energies.Ĥ|ψ⟩ = Eₙ|ψ⟩
QM6Fermionic antisymmetry: the wavefunction changes sign under exchange of two identical fermions (Pauli principle).ψ(r₁,r₂) = -ψ(r₂,r₁)
QM7Eigenvalue of the total spin operator, s(s+1)ħ², quantising spin angular momentum.Ŝ²|s,mₛ⟩ = s(s+1)ħ²|s,mₛ⟩
QM8Transmission coefficient for quantum tunnelling through a potential barrier (WKB form).T ∝ e⁻²∫√((2m/ħ²)(V-E))dx
QM9de Broglie relation linking a particle's wavelength to its momentum, λ = h/p.λ_dB = h/p
QP6Steinhardt bond-orientational order parameter Q6 for detecting crystalline ordering in many-body systems.Q_6 = \frac{1}{N_b}\sum_{i=1}^{N_b}\sum_{j\neq i} Y_6^m(\theta_{ij}, \phi_{ij})
TM0Imaginary-time evolution operator for thermal state preparation and path integral methods.φ × (1 - γ(1 - |φ|))
TQ0Quantum thermal time scale relating Planck's constant to thermal energy at temperature T.∫𝒟A e^(iS[A])
VX_QLQuantum vacuum fluctuation operator expressed via creation and annihilation operators.V_{QL} = \sqrt{\frac{\hbar}{2m\omega}}(a + a^\dagger)

Compute with one of these​

curl -sS -X POST https://zeqsdk.com/api/zeq/compute \
  -H "Authorization: Bearer $ZEQ_KEY" \
  -H "Content-Type: application/json" \
  -d '{"operators":["HG0"],"inputs":{}}'

The response carries the bare physics value, its unit and uncertainty, the generated master equation, and a signed envelope you can verify on any node.

See also​