RYORAKU vs Millennium Problems

RYORAKU is an interactive learning app that brings the invisible world of quantum mechanics to life. Watch particles spread like waves, energy come in discrete steps, and a spin that is decided only when measured. Developer: Fuji Bit Inc. Part of the "RAKU" physics series: see the quantum concepts taught in high school and university as moving pictures, not just formulas. 12 simulations in 3 categories: -- Waves & Particles - Wave function: A Gaussian wave packet moves and spreads, shown as its real part, imaginary part and probability density |ψ|². Change the wave number k, the width σ and the speed to see how a packet follows the Schrödinger equation. - Double slit: Particles reach the screen one at a time, yet interference fringes appear as they pile up. Change the slit spacing, slit width and wavelength and compare the fringes with the theory curve. - Uncertainty principle: Position and momentum distributions side by side. Squeeze the position spread Δx and the momentum spread Δp grows; their product Δx·Δp never drops below ℏ/2. - Quantum tunneling: A wave packet hits a potential barrier, and part of it gets through even when its energy is lower than the barrier. Change the barrier height and width and the particle energy to read the transmission T and reflection R. -- Quantized Energy - Particle in a box: Raise the quantum number n for more nodes and an energy that grows as n², or mix two states and watch the probability slosh back and forth. - Harmonic oscillator: Wave functions built from Hermite polynomials in a parabolic potential. See the evenly spaced levels E_n = ℏω(n+½) and the zero-point energy that stays above zero even at n = 0. - Hydrogen atom: Choose the quantum numbers n, l and m to change the shape of the s, p, d and f electron clouds, the radial probability and the energy E_n = −13.6/n² eV. - Hydrogen spectrum (new): Watch an electron drop from a higher level to a lower one and emit a photon whose wavelength matches the energy gap. Switch between the Lyman (ultraviolet), Balmer (visible) and Paschen (infrared) series to see why hydrogen glows in sharp lines, and why only four Balmer lines are visible. -- Spin & Entanglement - Stern-Gerlach: The historic experiment in which a beam of atoms splits in two in a non-uniform magnetic field. Change the input spin angle θ and the up/down split follows P(↑) = cos²(θ/2). - Entanglement: Alice and Bob measure entangled photon pairs and the correlations are recorded. Change the angles and the degree of entanglement to see quantum entanglement break the classical limit S = 2 of the Bell (CHSH) inequality. - Quantum walk: A quantum walk with a Hadamard coin next to a classical random walk. The quantum walk builds peaks at both edges through interference and spreads in proportion to the number of steps. - Qubit (new): A qubit shown as an arrow on the Bloch sphere. Change the drive strength Ω and the detuning Δ to watch Rabi oscillation as the state turns and the chances of ↑ and ↓ trade places. Press Measure and the state snaps to ↑ or ↓ according to its probability. Features: - Clean, modern design (switch between dark and light themes) - Home screen organized by category - Control panel with tips on what to try and a Play button - Info sheet with formulas and explanations for every simulation - In landscape and on iPad, the canvas and controls sit side by side - Supports larger text sizes - VoiceOver reads out each screen's results (transmission, measurement results and more) - Available in 73 languages, including English and Japanese (follows your device language) - No sign-in; start exploring right away Price and purchases: - Price: Free - In-app purchases: None - Subscriptions: None - Ads: Pangle banner ads at the bottom of the screen

Can you solve a problem worth $1,000,000? The Clay Mathematics Institute has offered a million-dollar prize for each of seven unsolved mathematical problems — and nobody has claimed them yet. Millennium Problems brings these mysteries to life for everyone, from curious 11-year-olds to doctoral researchers. SEVEN LEGENDARY PROBLEMS • P vs NP — Is finding as hard as checking? • Riemann Hypothesis — Where do prime numbers hide? • Yang-Mills & Mass Gap — Why do particles have mass? • Navier-Stokes — Can we predict turbulent fluids? • Hodge Conjecture — What shapes are built from algebra? • Birch & Swinnerton-Dyer — How many solutions does an equation have? • Poincaré Conjecture — The shape of the universe (SOLVED!) EVERY SKILL LEVEL Choose your grade — from Grade 6 to Graduate — and the content adapts instantly. The same problem is explained with simple analogies for a middle schooler and rigorous proofs for a PhD student. INTERACTIVE 3D VISUALISATIONS Every problem comes with a real-time Three.js visualisation. Watch elliptic curves rotate, Riemann zeros appear on the critical line, and fluid particle fields swirl — all rendered natively on your device. BUILT-IN QUIZZES Test your understanding with 5 questions per problem per grade level. Earn XP, track your streak, and collect badges as you explore. WORKS OFFLINE No account. No subscription. No ads. Everything runs locally on your device. OPTIONAL BIOMETRIC LOCK Protect your progress with Face ID, Touch ID, or fingerprint — entirely optional. Whether you're a student looking for a challenge, a teacher seeking engaging material, or a mathematician revisiting the classics, Millennium Problems meets you where you are. Clay Mathematics Institute · $7,000,000 in unclaimed prizes · Since 2000