Chemistry
Molecules, periodic table, reactions, pH scale, and titration
Hückel π-MO (Butadiene & Benzene)
Secular matrix H = αI + βA; eigen-energies and LCAO maps on the π skeleton.
MO Diagram: Homonuclear Diatomics (H₂–O₂)
σ/π ladder with Li–N vs O ordering; bond order from valence MOs; O₂ π* unpaired → paramagnetism.
Wigner Function (Coherent vs Squeezed)
Phase-space quasi-probability W(x, p) for a single-mode Gaussian quantum state: coherent |α⟩, displaced-squeezed D(α)S(ξ)|0⟩, and thermal. The 1σ ellipse rotates by half the squeeze phase θ/2 and shrinks below the vacuum floor along one quadrature — the basic picture of CV quantum optics.
Coherent State |α⟩ in a Harmonic Oscillator
Animated Gaussian wavefunction of a coherent state |α⟩ in a 1-D harmonic well: rigid σ = 1/√2 packet whose centroid traces the classical orbit ⟨x⟩(t) = √2|α|cos(ωt − φ_α). Side-by-side phase space, |ψ(x,t)|², and ⟨x⟩(t) trace.
WKB / Bohr–Sommerfeld Quantization
Semiclassical bound-state energies for arbitrary V(x): harmonic, quartic, Morse, double-well, asymmetric and square wells. Bisection on the action ∫√(2m(E−V))dx = (n+½)πℏ gives the WKB ladder; compare with the exact harmonic ladder ℏω(n+½).
Qubit Decoherence: Lindblad / T₁ T₂
Two-level Bloch master equation in the rotating frame with drive Ω, detuning Δ, T₁ relaxation and T₂ transverse decay. The Bloch vector spirals inside the sphere — the geometric picture of decoherence with live populations P(|0⟩), purity Tr ρ², and time traces of u_x, u_y, u_z.
Ramsey Fringes (Atomic Clock)
Separated-oscillating-fields sequence π/2 — τ — π/2 with detuning Δ and coherence T₂. Sweep τ or Δ to see P(|1⟩) = ½(1 − cosΔτ · e^{−τ/T₂}) — fringe period 2π/Δ, exponential T₂ envelope; the Bloch sphere animates each stage. Foundational to atomic clocks and Ramsey interferometry.
CHSH Bell Inequality Test
Monte-Carlo of the singlet state |Ψ⁻⟩ at four measurement angles (a, a′, b, b′): the running estimate Ŝ approaches the Tsirelson bound 2√2 ≈ 2.828 at the canonical Bell angles (0°, 90°, 45°, −45°), violating the local-realist limit |S| ≤ 2 — quantum entanglement made statistically visible.
Hong–Ou–Mandel Two-Photon Dip
Two indistinguishable photons enter opposite ports of a 50/50 beam splitter and bunch into the same output: coincidence probability P_c(δτ) = ½(1 − V·exp(−(δτ/τ_c)²)) (Gaussian) or Lorentzian. Drag the delay δτ to walk through the dip; live Monte-Carlo converges to the analytic curve. Visibility V = indistinguishability.
Kronig–Penney Bands & Brillouin Zone
Periodic δ-comb model of a 1-D crystal: cos(ka) = cos(qa) + (P/qa) sin(qa). Find allowed energy bands and forbidden gaps from the |·|≤1 corridor, then watch each band fold into the first Brillouin zone k ∈ ±π/a. Free-electron parabola overlaid for reference.
Phonon Dispersion: 1-D Mass–Spring Chain
Monoatomic ω = 2√(K/m)|sin(ka/2)| or diatomic acoustic/optical branches from alternating masses; ω(k) and group velocity v_g = dω/dk in the first Brillouin zone with animated lattice snapshot.
Debye vs Einstein Heat Capacity
Molar C_V(T): Debye integral → T³ law at low T and 3R Dulong–Petit at high T; compare with Einstein single-frequency model and toggle curves on one plot.
Anderson Localization (1D Tight-Binding)
Random onsite disorder W on a 1-D chain: diagonalize H, plot |ψ|² eigenstates, IPR vs energy, and localization length estimates (RMS and exponential fit).
Bloch Oscillations & Wannier–Stark Ladder
1-D tight-binding electron in uniform field E: semiclassical k(t), Bloch-periodic x(t), and finite-chain Wannier–Stark ladder with spacing ≈ eEa.
Graphene Tight-Binding Band Structure
Honeycomb π tight binding: E(k) along Γ–M–K–Γ, Dirac cones at K and K′ when Δ=0, gap 2Δ from sublattice stagger, and a Brillouin-zone energy map.
Quantum Hall Edge States & σ_xy Plateaus
Integer QHE: chiral edge channels and skipping orbits in a Hall bar, filled Landau levels vs μ, and quantized Hall conductance plateaus σ_xy = ν_f e²/h.
Drude–Sommerfeld Transport
Free-electron Drude model: scattering time τ, mobility μ, σ(ω), mean free path ℓ = v_Fτ, skin depth δ(ω), and DC Hall σ_xx(B), σ_xy(B).
2D Box: Eigenstates & Degeneracy
Particle in a 2-D rectangular infinite well: ψ_{n_x,n_y} ∝ sin(n_xπx/L_x)sin(n_yπy/L_y), E ∝ (n_x/L_x)² + (n_y/L_y)². Toggle a square box (L_x = L_y) to expose the (n_x, n_y) ↔ (n_y, n_x) accidental degeneracy and watch the doublets split as the box deforms.
Landau Levels in a Magnetic Field
Charged particle in a uniform B-field: equally-spaced Landau ladder E_n = ℏω_c(n+½) with cyclotron frequency ω_c = qB/m, magnetic length ℓ_B = √(ℏ/qB) and orbit radius r_n = ℓ_B√(2n+1). Animated cyclotron orbit + linear-in-B fan diagram; the n_B = qB/h degeneracy underlies the quantum Hall effect.
Maxwell–Boltzmann vs Eₐ
Translational energy density f(E); shaded fraction above activation energy; compare Arrhenius exp(−Eₐ/RT).
Water P–T Phase Diagram
Qualitative fusion, sublimation, vapor pressure up to critical point — probe labeled regions (pedagogical curves).
Close Packing FCC / BCC / HCP
Coordination numbers, maximal packing η, schematic ABC vs AB stacking beside a BCC cubic cell.
Chromatography Column
Partition chromatography cartoon: Gaussian bands separate as retention on the stationary phase differs.
Michaelis–Menten Kinetics
v vs [S] saturation and Lineweaver–Burk line from slope Km/Vmax and intercept 1/Vmax.
DNA Replication (Schematic)
Fork, leading vs lagging strand, Okazaki fragments — static labeled cartoon.
Polymer Random Coil
Lattice random walk R_g with Flory exponent ν slider (scaling hint vs Gaussian ν = ½).
Belousov–Zhabotinsky (Excitable)
Greenberg–Hastings excitable cellular automaton: target waves and rotating spirals as a qualitative BZ analogue.
Radioactive Decay & Chain
N(t) from half-life T₁/₂; optional parent → daughter → stable Bateman ODEs (RK4, normalized).
Le Châtelier Principle (Gas)
N₂ + 3H₂ ⇌ 2NH₃: T, V, add/remove species; Q vs K and relaxing mole bars.
VSEPR Molecular Shapes (3D)
Bonding vs lone pairs on a central atom: AXₙEₘ notation, electron geometry, and ball–lone-pair model.
Galvanic (Voltaic) Cell
Two half-cells, salt bridge, voltmeter; E°cell and Nernst E from ion concentrations (pedagogical E°).
Hess's Law (Enthalpy Paths)
Two-step vs direct ΔH on an enthalpy diagram; sum must match declared overall ΔH.
Collision Theory (2D Particles)
Hard-disk gas: elastic hits vs an activation speed threshold; T and Eₐ vs exp(−Eₐ/RT).
Crystal Field Splitting (Oct / Tet)
Δ_t = (4/9)Δ_o; HS vs LS d⁴–d⁷ from CFSE + pairing P; toy complex hue vs Δ_oct.
Unit Cell SC / BCC / FCC
Conventional cubic cells; yaw–pitch projection — lattice sites before basis detail.
Sequential Stern–Gerlach
Two SG devices: P(up on SG₂) = cos²(θ/2) or sin²(θ/2) after |±z⟩ filter.
Molecule Viewer (3D)
Common molecules in 3D. Rotate, zoom. Ball-and-stick models.
Periodic Table
Click element for properties, electron config, and uses.
Electron Configuration
Fill orbitals visually with Aufbau principle animation.
Balancing Equations
Interactive practice. Drag coefficients. Check balance.
pH Scale
Slider from 0 to 14 with common substances and color indicators.
Titration Simulator
Add base to acid. pH curve in real time. Find equivalence point.
Reaction Rate
Arrhenius k(T), order 0/1/2, [A](t) curves and particle tank.
Orbital Shapes (Schematic)
2D |ψ|² colormap for s-, p-, and d-like angular patterns (pedagogical, not HF).
Particle in a 1D Box
Superposition of n=1 and n=2: time evolution of |Ψ|² and ⟨E⟩ (model units).
Gaussian Wave Packet
Free spreading: σ(t) from dispersion with ℏ = m = 1; uncertainty intuition.
Quantum Well Eigenstates (Box & HO)
Stationary ψ_n and |ψ_n|² for infinite square well or harmonic oscillator; optional phase animation (ℏ = 1).
Rectangular Barrier Tunneling
Analytic transmission T(E) vs energy for a 1D barrier; tunneling below V₀ and resonances above (ℏ = m = 1).
Stern–Gerlach Beam (Cartoon)
Silver-like beam through an inhomogeneous B-field: atoms deflect to two detectors illustrating S_z = ±ℏ/2.
Bloch Sphere & Rabi Drive
Two-level Bloch vector with du/dt = u × ω, ω = (0, Ω, Δ); u_x–u_z view, Rabi flopping when Δ = 0.
Hydrogen |ψ|² (xz slice)
Coulomb hydrogen Z=1: |R_nl|² × |Y_lm|² colormap in the xz plane for textbook n,l states.
Hydrogen: Balmer / Lyman Lines
Rydberg wavelengths 1/λ = R_H (1/n_f² − 1/n_i²); Lyman, Balmer, and Paschen series on a linear spectrum strip.
Heisenberg Product σ_x σ_p
Gaussian position and momentum widths with ℏ = 1; momentum “excess” over the minimum-uncertainty floor.
β⁻ Decay: Electron Spectrum
Continuous kinetic-energy spectrum up to endpoint Q (allowed-decay phase space); missing energy carried by the antineutrino.
Nernst Equation
E = E° − (RT/nF) ln Q: sliders for E°, n, T, and reaction quotient.
Buffer Solution
Henderson–Hasselbalch vs strong acid: pH curve as H⁺ is added (mole model).
Acid Dissociation α(pH)
α = 1/(1+10^(pKa−pH)); half-equivalence at pH = pKa; curves for α and 1−α vs pH.
Born–Haber Cycle (NaCl & CaO)
Step ΔH° table and enthalpy ladder; lattice energy U = Σ(gas-ion steps) − ΔH_f°.
Frost Circle & Aromaticity (4n+2)
Inscribed n-gon on Hückel π energies; π count vs 4k+2 / 4k; Aufbau filling.
Henry's Law (Gas Solubility)
p = k_H c with k_H(T) from ΔH; c vs T at fixed p (ideal-dilute sketch).
Gray–Scott Patterns
Reaction–diffusion u,v; coral / mitosis / worms / spirals; D_u, D_v, Δt.
Gibbs Free Energy
ΔG = ΔH − TΔS; sign vs spontaneity at constant p,T (no Q or K).