{"query": "Madelung hydrodynamics — quantum mechanics as a probability", "count": 20, "results": [{"id": "card_n_810d857d886e", "title": "Madelung hydrodynamics — quantum mechanics as a probability fluid", "shelf": "science", "surface": "secular", "snippet": "Write ψ = √ρ·e^(iS/ħ) and the Schrödinger equation becomes fluid dynamics of the probability\ndensity ρ = |ψ|²: a continuity equation with velocity v = ∇S/m, plus an Euler equation carrying a\n'quantum "}, {"id": "card_c_41cd8b933961", "title": "Fluid probability dynamics — one continuity  ↔ Madelung hydrodynamics — quantum mechanics a", "shelf": "connections", "surface": null, "snippet": "quantum mechanics (Madelung: v = ∇S/m plus a quantum potential — the Schrödinger equation rewritten as fluid dynamics of |ψ|²)  — a concord the card itself states; mined + verified."}, {"id": "card_c_576d28632055", "title": "Madelung hydrodynamics instantiates Fluid probability dynamics", "shelf": "connections", "surface": null, "snippet": "Quantum mechanics rewritten as the fluid dynamics of |ψ|²."}, {"id": "card_n_8530c72a2201", "title": "Fluid probability dynamics — one continuity equation across physics, geometry, and ML", "shelf": "science", "surface": "secular", "snippet": "One equation wears many clothes. Probability is a conserved fluid: the continuity equation\ndρ/dt + ∇·J = 0 (current J = ρv) says density is never created or destroyed — it only flows. The\nsame skeleto"}, {"id": "card_src_word_quantum_mechanics", "title": "quantum mechanics", "shelf": "dictionary", "surface": "secular", "snippet": "quantum mechanics: (noun) the branch of quantum physics that accounts for matter at the atomic level; an extension of statistical mechanics based on quantum theory (especially the Pauli exclusion prin"}, {"id": "card_src_word_wave_mechanics", "title": "wave mechanics", "shelf": "dictionary", "surface": "secular", "snippet": "wave mechanics: (noun) the modern form of quantum theory; an extension of quantum mechanics based on Schrodinger's equation; atomic events are explained as interactions between particle waves"}, {"id": "card_c_b0dd29a6dbce", "title": "The periodic table — the map of the elements ↔ Madelung hydrodynamics — quantum mechanics a", "shelf": "connections", "surface": null, "snippet": "its periodicity is the fingerprint of the quantum shells beneath  — a concord the card itself states; mined + verified."}, {"id": "card_n_041763374eca", "title": "Fokker–Planck & Liouville — probability flow in statistical mechanics", "shelf": "science", "surface": "secular", "snippet": "The stochastic instantiation of probability-as-fluid. Fokker–Planck sets the current\nJ = μρ − D∇ρ (drift + diffusion); its steady state is the Boltzmann distribution ρ ∝ e^(−U/kT),\nwhich makes the cur"}, {"id": "card_c_c1d4f07c75cd", "title": "Fokker–Planck & Liouville — probability flow ↔ Fluid probability dynamics — one continuity ", "shelf": "connections", "surface": null, "snippet": "The stochastic instantiation of probability-as-fluid. ... In Hamiltonian phase space, Liouville's theorem gives dρ/dt = 0 — the probability fluid is incompressible.  — a concord the card itself states"}, {"id": "card_c_3cc9f3e8e74f", "title": "Fokker–Planck & Liouville instantiates Fluid probability dynamics", "shelf": "connections", "surface": null, "snippet": "Statistical mechanics is the stochastic instance of the probability-continuity equation."}, {"id": "card_src_word_quantum_field_theory", "title": "quantum field theory", "shelf": "dictionary", "surface": "secular", "snippet": "quantum field theory: (noun) the branch of quantum physics that is concerned with the theory of fields; it was motivated by the question of how an atom radiates light as its electrons jump from excite"}, {"id": "card_theory_quantum_mechanics", "title": "Quantum mechanics (Schrödinger equation)", "shelf": "theories", "surface": "secular", "snippet": "Quantum mechanics (Schrödinger equation) — an engine domain that can touch it: physics. Calibration: partial — specific relations verify; the theory as a whole is not a sealable computation. energy le"}, {"id": "card_c_e655b3fcef54", "title": "Optimal transport instantiates Fluid probability dynamics", "shelf": "connections", "surface": null, "snippet": "The geometry of moving probability — Fokker–Planck as a Wasserstein gradient flow."}, {"id": "card_c_d7daafda1482", "title": "Fluid probability dynamics — one continuity  ↔ Diffusion models — the probability-flow ODE ", "shelf": "connections", "surface": null, "snippet": "machine learning (the probability-flow ODE under diffusion models)  — a concord the card itself states; mined + verified."}, {"id": "card_src_book_62227", "title": "Bicycling for Ladies The Common Sense of Bicycling; with Hints as to the Art of Wheeling—Advice to Beginners—Dress—Care of the Bicycle—Mechanics—Training—Exercise, etc., etc. — Mar", "shelf": "gutenberg", "surface": "secular", "snippet": "Bicycling for Ladies The Common Sense of Bicycling; with Hints as to the Art of Wheeling—Advice to Beginners—Dress—Care of the Bicycle—Mechanics—Training—Exercise, etc., etc., by Maria E. Ward. Subjec"}, {"id": "card_src_word_max_born", "title": "max born", "shelf": "dictionary", "surface": "secular", "snippet": "max born: (noun) British nuclear physicist (born in Germany) honored for his contributions to quantum mechanics (1882-1970) — syn: Born"}, {"id": "card_src_word_roald_hoffmann", "title": "roald hoffmann", "shelf": "dictionary", "surface": "secular", "snippet": "roald hoffmann: (noun) United States chemist (born in Poland) who used quantum mechanics to understand chemical reactions (born in 1937) — syn: Hoffmann"}, {"id": "card_src_word_dirac", "title": "dirac", "shelf": "dictionary", "surface": "secular", "snippet": "dirac: (noun) English theoretical physicist who applied relativity theory to quantum mechanics and predicted the existence of antimatter and the positron (1902-1984) — syn: Paul Adrien Maurice Dirac"}, {"id": "card_src_word_paul_adrien_maurice_dirac", "title": "paul adrien maurice dirac", "shelf": "dictionary", "surface": "secular", "snippet": "paul adrien maurice dirac: (noun) English theoretical physicist who applied relativity theory to quantum mechanics and predicted the existence of antimatter and the positron (1902-1984) — syn: Dirac"}, {"id": "card_src_word_born", "title": "born", "shelf": "dictionary", "surface": "secular", "snippet": "born: (noun) British nuclear physicist (born in Germany) honored for his contributions to quantum mechanics (1882-1970) — syn: Max Born · (adjective) brought into existence · (adjective) being talente"}]}