§ Glossary

A planar (or conical) surface where the wavefunction crosses zero due to the angular factor Y_ℓm. An (n, ℓ) state has ℓ of them.

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ℓ ∈ {0, 1, …, n−1}. Long alias for the l entry — sets the orbital's angular shape.

A potential V(x) higher than the particle's energy E over a finite region. Classical particles bounce; quantum particles partly transmit (tunnel) and partly reflect.

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The unit sphere on which every spin-½ pure state sits. The poles are |↑⟩ and |↓⟩; the equator carries the equal superpositions with varying phase.

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a₀ ≈ 0.529 Å. The natural length scale of atomic physics — and the peak of the hydrogenic 1s radial probability density.

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|ψ(x)|² is the probability density for position; |⟨φ|ψ⟩|² is the probability of observing the system in state |φ⟩. The interpretive bridge between the Schrödinger equation and what a detector clicks.

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The post-measurement reduction of ψ to the eigenvector matching the outcome. Whether this is a real physical process or a bookkeeping update is the heart of the interpretation debate.

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[Â, B̂] = ÂB̂ − B̂Â. Zero iff the operators share a common eigenbasis (simultaneously measurable); nonzero forces an uncertainty relation.

Volumetric front-to-back composite of |ψ|² along each ray. Brighter = higher probability density. Soft and cloud-like.

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Bra-ket shorthand: |ψ⟩ is a state vector (ket), ⟨φ| its dual (bra), ⟨φ|ψ⟩ their inner product, |ψ⟩⟨φ| an outer-product operator.

Two attractive basins separated by a barrier. The ground state splits into a symmetric/antisymmetric pair — the prototype for tunneling, ammonia inversion, and the qubit.

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A state for which an operator returns the same state times a number — like ψ_nlm for the hydrogen Hamiltonian, which always returns its eigenenergy E_n.

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A multi-particle state that can't be written as a product |ψ_A⟩ ⊗ |ψ_B⟩. Measurements on the parts are correlated in ways no local hidden-variable theory reproduces.

⟨A⟩ — the statistical mean of operator A measured on a state. For ⟨r⟩ in a hydrogenic ψ_nlm: (n²/Z)·(1 + ½·(1 − ℓ(ℓ+1)/n²)) Bohr.

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Ĥ — the energy operator, kinetic plus potential: Ĥ = −(ℏ²/2m)∇² + V(x). Its eigenvalues are the allowed energies; diagonalising it solves the system.

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V(x) = ½ m ω² x². Eigenenergies are evenly spaced, Eₙ = ℏω(n + ½); eigenfunctions are Hermite polynomials × Gaussian. The textbook potential because it solves cleanly.

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A self-consistent mean-field method that handles electron-electron repulsion approximately. The screened-Z option in the viewer comes from a small HF run; see the about page.

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An operator equal to its own adjoint, † = Â. Eigenvalues are real and eigenvectors are orthogonal — exactly the structure observables need.

The complex inner-product vector space in which state vectors live. For a single particle in 3D it's L²(ℝ³); for spin-½ it's just ℂ².

A one-electron model atom: nucleus of charge Z, no electron-electron repulsion. Real multi-electron atoms differ via screening; here it's the baseline.

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Two probability amplitudes that share the same outcome add as complex numbers, so |a + b|² ≠ |a|² + |b|² in general — the source of double-slit fringes and quantum-walk patterns.

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The 3-D surface where |ψ|² equals a chosen threshold — a 'skin' enclosing the region of high probability. Adjust the threshold to peel layers.

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Angular-momentum quantum number ℓ. Sets the orbital's shape: ℓ=0 sphere (s), ℓ=1 dumbbell (p), ℓ=2 cloverleaf (d), ℓ=3 (f). Ranges 0 to n−1.

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The radial polynomial in the hydrogenic eigenfunction R_nℓ(r). Generates the n − ℓ − 1 radial nodes via its zeros.

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A spin in a magnetic field B precesses around B at ω = γB. On the Bloch sphere: a tilted axis traces a cone at constant polar angle.

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Magnetic quantum number m (often mₗ). Sets the orbital's orientation around the z-axis; integer values from −ℓ to +ℓ.

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m ∈ {−ℓ, …, +ℓ}. Long alias for the m entry — sets orientation around the z-axis.

Projects the state onto an eigenvector of the measured observable; the outcome is the corresponding eigenvalue, with probability |⟨eigvec|ψ⟩|². Irreversible in standard QM.

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Principal quantum number n. Sets the shell, the energy E_n ∝ −Z²/n², and roughly how far the electron sits from the nucleus.

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The everyday name for an angular node when it really is flat — for example, the xy-plane that separates the two lobes of 2p_z.

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∫ |ψ|² dx = 1. The state's total probability is 1; if a sim drifts from this, the integrator is leaking probability.

A linear map on the Hilbert space of states. Observables (position, momentum, energy, spin) are Hermitian operators; eigenvalues are the possible measurement outcomes.

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No two electrons share the same complete set of quantum numbers. With spin-½, each (n, ℓ, m) orbital holds at most two electrons — the rule that builds the periodic table.

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σ_x, σ_y, σ_z — the three 2×2 Hermitian, traceless, unitary generators of spin-½ rotations. Anticommute pairwise, square to the identity.

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The complex argument of ψ. Global phase is unobservable; relative phase between superposed components is what interference and Bloch-sphere rotations actually measure.

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n ∈ {1, 2, 3, …}. Long alias for the n entry — sets shell and energy.

j = (ℏ/m) Im(ψ* ∇ψ). The flux of |ψ|² — what flows through a slit or across a barrier in a scattering setup.

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A two-level quantum system — any one is isomorphic to a spin-½. The computational basis |0⟩, |1⟩ is just a relabeling of the Bloch poles.

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Mean radial distance for the state — the dashed line on the radial-probability inset. NOT the most-probable radius (those agree only for the 1s ground state).

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A spherical surface where the wavefunction crosses zero — the electron is exactly NOT found there. An (n, ℓ) state has n−ℓ−1 of them.

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Rendering technique: step a ray through a volume, accumulating colour and opacity at each sample. The viewer raymarches the precomputed |ψ|² cubes at video rate.

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R(E) = 1 − T(E) (no absorption). The fraction of probability that bounces back from a barrier or step.

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iℏ ∂ψ/∂t = Ĥψ. The first-order-in-time PDE that propagates ψ forward; the time-independent form Ĥψ = Eψ extracts the stationary states (eigenstates of energy).

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Inner electrons partly shield outer ones from the nuclear charge, so each outer electron sees an effective Z_eff < Z. Switch to the HF model (Z=1..18) to see it.

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Atom-mode trick: weight the density by r² so the textbook radial probability 4πr²·ρ(r) lights up in 3D. Closed-shell atoms reveal their nested-shell structure under it.

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A clipping half-space along X, Y, or Z. Cuts the volume open so you can see its interior — especially useful with the shell-reveal atom view.

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V(x) = −1/√(x² + a²). A 1D stand-in for the 1/r potential that keeps the energy finite at the origin; used to study atomic-like spectra without numerical singularity.

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Y_ℓm(θ, φ) — the angular part of every hydrogenic orbital. Real combinations give the familiar p_x / p_y / p_z and d_xy / d_yz shapes.

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An intrinsic angular momentum with no classical analogue. For an electron, spin-½: two basis states |↑⟩, |↓⟩, eigenvalues ±ℏ/2 of Ŝ_z.

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Time-evolves ψ on a grid by alternating kinetic and potential half-steps — kinetic via FFT (diagonal in momentum), potential pointwise (diagonal in position). The 2D sandbox uses this.

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An infinite or finite box potential. The infinite well has Eₙ ∝ n²; bound states pile up against the walls and zero out at the boundary.

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An energy eigenstate. Its |ψ|² is time-independent — only the phase rotates as e^{−iEt/ℏ}. Building blocks for any time-evolving state.

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Any linear combination c₁|ψ₁⟩ + c₂|ψ₂⟩ of states is itself a valid state. Interference patterns and quantum parallelism both descend from this one fact.

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T(E) — the fraction of a wave packet's probability current that makes it through a barrier. Plotted as the T(E) curve in the 1D scattering sandbox.

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A particle penetrates a classically forbidden region (V > E) with a small but nonzero amplitude. The exponential tail of ψ inside the barrier is what makes scanning-tunneling microscopy work.

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For complementary observables Â, B̂: Δ Â · Δ B̂ ≥ ½|⟨[Â, B̂]⟩|. The position-momentum case Δx · Δp ≥ ℏ/2 is the famous one, but spin components obey it too.

The sum Σ_m |Y_ℓm|² is rotation-invariant (a constant). So a fully-filled subshell contributes a spherically symmetric electron density — that's why closed-shell atoms 'look round'.

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A localised superposition of plane waves, typically Gaussian. Has finite position and momentum spreads obeying Δx · Δp ≥ ℏ/2; the canonical thing to send through a barrier or slit.

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ψ(x) — the complex-valued amplitude whose modulus squared |ψ|² is the probability density of finding the particle at x. The whole of nonrelativistic QM is bookkeeping for this object.