Educational Resources

Change the slit width and the particle mass and speed, then launch particles individually or 10,000 at a time to accumulate their landing positions. Watch the histogram approach the theoretical Fraunhofer diffraction curve ${|\psi (x)|}^2$. Keeping the display range fixed while increasing the mass from an electron to a baseball makes the distribution collapse to a single point, approaching the classical picture.

Adjust a (half the width of the central barrier) and b (the width of each slit), then click “▶ Fire 1 particle” to send one particle at a time or “Fire 10,000” to accumulate landing points. For the same particle and screen distance, compare the on-screen readouts to see that the center-to-center distance $d=2a+b$ determines the fringe spacing, while b determines the width of the overall pattern. In the default view, each particle beyond the slits is depicted as a wave spreading from both openings and is recorded as a single point only on the screen.

Adjust the barrier height and thickness. Panel ① redraws the transmission coefficient T and reflection coefficient R as functions of incident energy, marking the values at your chosen kinetic energy E. See that T remains nonzero even when $E<V_0$, that in this regime a slight increase in thickness changes T by orders of magnitude (the principle behind scanning tunneling microscopy), and that resonant transmission appears above the barrier. In ②, propagate a wave packet and compare the probabilities remaining on each side with theoretical values averaged over the wave packet.

Move the temperature T slider from 3 K to 30,000 K, or select red-hot iron or a star, to see the blackbody spectrum and its perceived color. As temperature rises, the peak shifts toward shorter wavelengths (Wien's displacement law). Explore how a star's color reveals its surface temperature.

Reduce the mass from a baseball to an electron and then to an even lighter hypothetical particle. Set the launch speed and angle. In a uniform gravitational field, the mean position follows the same classical parabola regardless of mass; only the surrounding spread $\sigma (t)$ changes. As the mass decreases, the trajectory disappears into a cloud and the landing positions become more scattered.

Choose a system ranging from a vibrating molecule to a tabletop spring. In ①, increase n to 20 and watch the distribution approach the classical probability density. The energies are discrete, $E_n=(n+\frac{1}{2})\hslash \omega$, and in ②, the lowest energy never reaches zero even as the temperature decreases. In ③, points sampled from the probability distribution oscillate while remaining blurred. In ④, add or remove one quantum at a time and follow the changing number of wave-function nodes. In ⑤, compare the same operations for bosons, which can share a state, and fermions, with at most one particle per state.

Select an orbital from 1s to 4f and a magnetic quantum number $m$. In ①, drag to rotate points sampled from the probability density (each point represents one measurement), or view a cross section to reveal shells and nodes. In ②, compare the radial distribution $P(r)=r^2{|R|}^2$ with angular cross sections to explore how the number of nodes and the likely distance of the electron vary between orbitals.

In ①, click to fill 2p, 3d, or 4f levels with electrons and build the total energy using an orbital-independent Coulomb integral $U_0$ and exchange integral $J$. With no coordination, observe Hund's first rule (maximum S). For octahedral coordination with 4–7 electrons in 3d orbitals, adjust the ligand-field strength $\Delta$ to explore the crossover between high-spin and low-spin states. In ②, model ligands as point charges and select octahedral, tetrahedral, or tetragonally distorted arrangements—an introduction to the Jahn–Teller effect—while comparing a rotatable diagram of the arrangement with the energy-level diagram.

Select a diagram and click its straight lines, wavy lines, and loops to compare the mathematical factors with animations of an electron sea (①–②). In ③, change the approximation order and temperature to follow how the imaginary part of the self-energy produces spectral broadening and decay of the survival amplitude. In ④, switch to phonons: vary the potential depth and temperature, toggle the quartic loop and the two cubic diagrams, and compare frequency shifts with the exact Morse-type solution. Diagrams record how lines are connected; they are not electron trajectories.

Choose a particle and vary its momentum from 10⁻³mc to 5mc. In ①, see where the Schrödinger parabola departs from the Klein–Gordon and Dirac hyperbola $E=\pm \sqrt{p^2{c}^2+m^2{c}^4}$, whose lower branch represents antiparticles. Vary the box width in ②, electron density in ③, and atomic number from 1 to 120 in ④. Explore how replacing the dispersion relation and boundary conditions changes the energy levels in a box, the pressure of an electron gas, and the electron distribution near the nucleus.

Click the glowing particles to descend through an atom to the quark scale (①). In ②, vary distance from 10⁻¹⁹ to 10⁻⁹ m and see how the relative strengths of the four forces trade places. In ③, examine how electrons and quarks, whose intrinsic sizes have only upper bounds, can fluctuate over vastly larger distances, while the comparison reverses for atoms in materials. In ④, replace the exchanged photon with a massive particle to see the interaction range shrink to $\hslash /mc$.

Choose SO(3), SU(2), or Sp(2,ℝ), vary its parameters, and monitor the invariants. Deliberately step outside the group and watch those values change: each name declares which quantities its transformations preserve. Rotate through 720° to see that an orientation returns after one turn but a spinor requires two. Continue through raising and lowering operators, planar SU(3) diagrams, the classification of Lie algebras, and grand unification grouping the 16 states of one generation.

In the upper simulator, choose among 12 elements and vary amplitude (Higgs) and phase (Nambu–Goldstone) around the rim of the “Mexican hat” Ginzburg–Landau free energy. Raise the temperature to see the rim shrink into a bowl and explore what sets the transition temperature (①–③). In ④, a lattice order parameter produces the same free-energy shape, with imaginary-frequency phonons corresponding to the negative curvature at the center. In the lower simulator, heavier ions leave a longer wake that attracts distant electrons; stronger coupling changes pairs from bubbles larger than the electron spacing into small molecules.

Choose a gas, temperature, pressure, and volume, then press “Run” and the molecules begin flying around in the box. In ①, the pressure calculated solely by summing impulses on the walls, without using $nRT/V$, nearly matches the ideal-gas value. Even after you press “Make all speeds equal,” collisions alone restore a Maxwell distribution. In ②, choose a fixed wall or a piston and heat the gas. Compare the heat-input and work bars to see that the extra heat needed with the piston corresponds to expansion work ($C_P-C_V=R$).

Set the hot and cold reservoir temperatures, working gas, and expansion ratio, then run the cycle. In ①, points travel around two diagrams in step with the piston. The enclosed area of the P–V diagram gives the work per cycle, while the T–S diagram forms a rectangle. In ②, follow the heat and work for each stage in a table and verify that the efficiency $\eta =1-\frac{T_C}{T_H}$ depends only on the two reservoir temperatures and does not change when the working gas is replaced.

Choose one of eight gases, set its inlet temperature and pressure drop, and pass it through an insulated restriction. In ①, because an ideal gas shows no temperature change, calculate with a van der Waals real gas and confirm that this constant-enthalpy process cools the gas inside the inversion curve on the T–p diagram, while He and H₂ lie outside the curve at room temperature and warm up. In ②, use the same throttling process as an air conditioner's expansion valve, choose among six refrigerants, and examine p–h diagrams and COP values for cooling and heating.

Place different gases on each side at equal temperature and pressure, then remove the partition. In ①, the entropy obtained by counting the molecules as they mix approaches $\Delta S=-nR\Sigma x\ln x$. With identical gases on both sides, only the color-based counting increases, while the thermodynamic $\Delta S$ is zero: Gibbs' paradox. In ②, verify that the direction of mixing coincides with the direction in which the Helmholtz free energy $F$ decreases (at constant temperature and volume) or the Gibbs free energy $G$ decreases (at constant temperature and pressure).

Vary the external pressure from 1 Pa to 10 bar. In ①, plot $G=H-TS$ against temperature for ice, water, and water vapor. The crossings where the lowest curve changes determine the melting and boiling points. Below the triple-point pressure, the liquid curve is never the lowest at any temperature, so ice sublimates directly. Follow the competition in which lower H wins at low temperature and larger S wins at high temperature. In ②, collect the crossings at different pressures to construct the phase diagram.

In ①, increase the number of atoms $N$ from 2 to 60 and click energy levels to see the shapes of their orbitals. The levels crowd into a fixed energy range and form a band. In ②–③, close the chain into a ring: all allowed $k$ values lie on one dispersion curve. As more atoms are added, the points become denser and the distribution concentrated near the band edges approaches the theoretical curve. In ④, alternate atoms or bonds to open a gap, splitting a half-filled band into a full band and an empty band to produce an insulator. In ⑤, choose an orbital from s to f and the phase pattern, reciprocal-lattice points, and band diagram respond together; differences in the way the orbitals overlap determine the bandwidths and splittings.

Choose a material, thickness, and initial temperature. In ①, switch between a fixed surface temperature and convective cooling to follow the temperature profile and mode amplitudes together. Fine temperature variations decay fastest, leaving a single broad peak. With radiation disabled, the series and numerical solutions coincide; convection changes the eigenvalue problem itself. In ②, surround the slab with air, water, or refractory brick and solve the surrounding region too. You will find that the interface temperature immediately after contact is determined not by thermal diffusivity but by thermal effusivity (the reason metal feels cold to the touch).

Choose a material and temperature. View the nonzero vibration amplitude of a two-dimensional crystal even at 0 K in ①, compare classical (Dulong–Petit), Debye, and Einstein lattice specific heats in ②, and inspect the Boltzmann distribution on an energy-level ladder in ③. Specific heat is nothing other than how much the energy can fluctuate. Modes with $ħ\omega \gg k_{\text{B}}T$ remain in their lowest level and cannot absorb appreciable heat. Whether room temperature counts as high temperature depends on its ratio to ${\Theta }_{\text{D}}$.

Choose a material. In ①, rotate the supercell while increasing temperature to see atomic fluctuations and lattice expansion. Ideal springs leave the mean position unchanged, so expansion requires anharmonicity. Summarizing it with the Grüneisen parameter in the quasiharmonic approximation gives ${\alpha }_V=\gamma C_V/(B_{\text{T}}V)$. In ②, compare 38 cubic crystals and change the horizontal axis to see that the magnitude of positive thermal expansion is approximately inversely proportional to the bulk modulus. In ③, overlay the temperature dependence of expansion, stiffness, and specific heat for two materials.

In ①, select crystal planes in CsCl, SrTiO₃, or NaCl and see waves from adjacent planes reinforce at $2d\sin \theta =\lambda$, while the atomic layers within one period determine peak intensity. In ②, draw the reciprocal lattice and Ewald sphere to confirm that the Laue condition says the same thing as Bragg's condition. In ③, draw the structure factor as a sum of arrows in the complex plane and follow the systematic absences, in which reflections cancel and vanish, as in face-centered cubic lattices. In ④, keep the lattice parameters fixed and change the contents of the unit cell to verify that the Bravais lattice alone sets the possible peak positions in 2θ. In ⑤, calculate powder XRD patterns for 27 real crystal structures. In ⑥, estimate crystallite sizes from peak widths using the Scherrer equation.

In ①, see calcination, the step before sintering, in which two powders in contact mix by atomic diffusion to form an ordered compound or a solid solution. From ② onward, turn to sintering itself and confirm that its driving force is the reduction of surface energy. In ③, compare neck growth in a two-sphere model on log–log plots: the exponent $n$ in ${(x/a)}^n\propto t$ is 7 for surface diffusion, 6 for grain-boundary diffusion, 5 for volume diffusion, and 3 for evaporation–condensation. In ④, see that whether matter comes from the particle surface or the grain boundary determines whether shrinkage occurs. In ⑤–⑥, plot densification and see it stop when growing grains leave pores behind. In ⑦, compare how effective each of four approaches is: pressure, a liquid phase, powder design, and additives.

Choose an isomorphous, eutectic, or peritectic system. In ①, drag across the phase diagram to read the phases, microstructure, and phase fractions calculated by the lever rule in two-phase regions. In ②, use Gibbs free-energy curves and their common tangent to trace the equilibrium compositions and construct the phase diagram. In ③, cool a melt and compare equilibrium cooling with Scheil-type solidification without solid-state diffusion to see microsegregation and nonequilibrium eutectics in schematic microstructure diagrams. In ④, compare glass formation on rapid quenching with crystallization on slow cooling. In ⑤, use schematic atomic arrangements to see how a glass or solid solution that is homogeneous at high temperature separates into two phases by spinodal decomposition on cooling.

Ceramics can be hard and strong yet fracture suddenly with little deformation. In ①, compare the elongation and failure of metal and ceramic specimens in tensile tests. In ②, explore stress concentration near holes and cracks, and the strength ${\sigma }_{\text{f}}=\sqrt{2E{\gamma }_{\text{s}}/(\pi a)}$ predicted by Griffith's energy balance. In ③, use a schematic of the atoms at the crack tip to see that the tip blunts in metals, where dislocations can move, but bonds break and the crack runs in ceramics, where they cannot. Use fracture toughness K_IC to calculate the tolerable crack size for each material. In ④, break specimens one by one to see strength scatter follow a Weibull distribution. In ⑤, see how strength differs with specimen volume and test method. In ⑥, press a diamond indenter into a sample and measure the residual indentation diagonals yourself to calculate Vickers hardness, then check against the automatic measurement. In ⑦, inspect temperature and stress distributions during thermal shock.

In ①, see the origin of the Seebeck effect, in which a temperature difference produces a voltage: charged particles on the hot side diffuse toward the cold side, leaving a charge imbalance. In ②, use a single parabolic band model to calculate the Seebeck coefficient S, electrical conductivity σ, and electronic thermal conductivity κ_e versus carrier concentration. Their competing effects create a peak in the figure of merit $ZT=S^2\sigma T/\kappa$, typically at 10¹⁹–10²¹ cm⁻³. In ③, see that the quality factor B alone determines the peak height, while the effective mass shifts its position. Explore reducing lattice thermal conductivity (PGEC) in ④, conversion efficiency and the cooling coefficient of performance (COP) in ⑤, and applications of real materials in ⑥.

For solid-state synthesis, add ethanol to powders, grind, dry, and calcine. For solution synthesis (coprecipitation), add solutions dropwise, precipitate, and collect the product. In CVD, adjust substrate temperature, pressure, and precursor gases; in PVD, switch between vacuum evaporation and sputtering. Compare product layers growing at particle contacts, mixing and precipitation of ions in solution, chemical reactions and gaseous by-products at the substrate surface, and the attachment of ejected atoms and film growth. Linked apparatus diagrams, schematics, and graphs show how the state of the starting materials, transport paths, and locations of reaction or deposition differ between methods and determine the resulting powders or films.

Weigh Y₂O₃, BaCO₃, and CuO, mix with ethanol, then follow one sample through calcination, regrinding, pressing, sintering, oxygenation, polishing, bulk-density measurement, and four-probe resistance measurement. Track phase changes, densification, and oxygen deficiency δ. Cool your sample or a prepared sample (melt-grown YBCO bulk, GdBCO bulk, or sintered Bi-2223) with liquid nitrogen and place it over a magnet. Using flux diagrams and force plots, compare zero-field cooling with field cooling, and see how pressing the sample toward the magnet, moving it sideways, and warming it change the behavior. Being below T_c does not guarantee identical levitation: oxygen content, microstructure, current-loop size, and cooling history all matter.

Pour iron chloride and barium chloride solutions, with a chosen starting Fe/Ba ratio, into sodium hydroxide and sodium carbonate solution. Wash the precipitate with warm water, vacuum-filter, dry, calcine in an electric furnace, then press, sinter, and magnetize the powder. Track pH, precipitation yield, components lost to the supernatant and filtrate, changes in phase fractions and mass during calcination, and apparent remanent magnetization measured with a balance. Schematics show ions, particles, grains, pores, and magnetization directions. The precipitate is a precursor made of hydrous iron oxide and BaCO₃; BaFe₁₂O₁₉ forms only during calcination. Explore how an off-stoichiometric feed, dissolution during washing, or insufficient calcination temperature affects the product and magnet strength.

Choose the SiO₂–Al₂O₃–MgO–B₂O₃–Na₂O–K₂O composition in mol% or mass%. Weigh the actual raw materials (silica sand, carbonates, and boric acid), mix in a mortar, and melt in an electric furnace. Then choose water quenching, pressing between metal plates, furnace cooling, or another cooling method. Follow CO₂ bubbles from carbonate decomposition, bubble removal and homogenization, the roles of Al and B in the network, and cooling curves overlaid on CCT and TTT diagrams. Glass formation depends on both cooling rate and composition; not every mixture forms a glass.

Choose a powder, water-to-powder ratio, and temperature, then add the powder, mix, and advance time. With calcined gypsum (plaster of Paris), the solution becomes supersaturated with respect to gypsum dihydrate and needle-shaped crystals interlock. With Portland cement, C–S–H, calcium hydroxide, and ettringite grow from particle surfaces and connect them. Follow reaction progress, heat release, and porosity. Setting occurs because water-containing hydrates precipitate and connect, not because the water evaporates. More mixing water leaves more pores.

In ①, choose among ten functions with different convergence behavior and increase the order to see how far the accurate range extends and where it stops. In ②, move the evaluation point and numerically locate the c whose existence is all that the theorem asserts. Compare the actual error with the Lagrange bound and see how singularities in the complex plane determine the radius of convergence. In ③, use a simple pendulum and a Lennard-Jones potential to explore why an expansion requires specifying what the small quantity is small relative to.

In ①, use the two coefficients of the molecular orbital $\Psi =c_{\text{A}}{\psi }_{\text{A}}+c_{\text{B}}{\psi }_{\text{B}}$ as coordinates. Go once around the ellipse drawn by the normalization constraint and find the lowest-energy point. In ②, vary the multiplier $\epsilon$ to see that a nontrivial (nonvanishing) wave-function solution appears only at those values of $\epsilon$ for which the determinant vanishes. In ③, use presets or sliders to separate ${\alpha }_{\text{A}}$ and ${\alpha }_{\text{B}}$, and follow the shift from covalent to ionic bonding in the energy levels and wave function.

Move two vectors in ① and a third vector in ②. Confirm that the parallelogram area or parallelepiped volume equals |det|, checking against base × height or base area × height. Use the buttons to make two vectors coincide or to drop the third into the plane of the other two, and the determinant becomes zero. In ③, select an example and change the third row of the coefficient matrix to see the solution set expand from the origin to a line and then a plane as the rank decreases.

Choose an orbital and adjust the basis-mixing or rotation controls to find the angle at which the red off-diagonal elements vanish. In ①, switch 2p, 3d, and 4f orbitals between real and complex forms to compare zero and nonzero probability currents. In ②, see the x-directed spin ${\sigma }_x$ become simply up or down in a new basis. In ③, see that the displacements of two spring-coupled masses remain unchanged while the new components separate into two independent simple harmonic oscillations.

In ①, label the outcomes by dividing 0–6 into n equal intervals, then increase the number of faces from 6 to 1,200. Each outcome's probability tends to zero, but the probability density, obtained by dividing by the interval width, stays constant. In ②, choose a two-peaked distribution to see that the expectation value lies at neither peak. In ③, increase the subdivisions to 200 and watch the discrete sum approach $⟨x⟩=\int x{|\psi |}^{2}dx$. In ④, vary the mixture and make measurements: only the two energy levels appear as outcomes, while their average lies between them.

In ①, choose among six waveforms and increase N to 60. The broad shape emerges first, while Gibbs oscillations persist near discontinuities. In ②, see orthogonality as cancellation of the signed area under the product of two basis functions. In ③, evolve an initial profile with the heat and wave equations: each mode evolves independently, and finer modes decay faster in heat conduction. In ④, choose a metal and vary the electron density to explore Thomas–Fermi screening. Because differentiation becomes multiplication by wave number, an equation including the electronic response can be solved by division.

Increase the damping γ in ① to move from underdamped through critically damped to overdamped motion. In ②, see that the transition occurs when two roots slide along a circle in the complex plane and meet on the real axis. Apply a driving force in ③: with damping, only oscillations at the driving frequency survive at long times, regardless of the initial conditions. In ④, vary Q from 1 to 100 to see that the same number controls both the height and narrowness of the resonance.