Standing Waves · Orbital Shapes · Quantum Numbers · Electron Configuration
Only discrete standing-wave modes persist on a plucked string. Fixed endpoints require an integer number of half-wavelengths:
Energy is therefore quantised — only specific values are permitted. The integer n is a mode index.
An electron bound to a nucleus obeys a 3-D wave equation. Atomic orbitals are the 3-D standing-wave solutions confined by the Coulomb potential:
The quantum numbers n, ℓ, mℓ are the 3-D mode indices. Each unique triple defines one orbital "mode shape."
Blue — positive phase (ψ > 0)
Orange — negative phase (ψ < 0)
Axes: x (red) · y (green) · z (blue)
Cr & Cu anomalies included · toggle for noble gas notation
Aufbau Lowest-energy subshells first: 1s → 2s → 2p → 3s → 3p → 4s → 3d → 4p
Pauli Max 2 electrons per orbital, opposite spins (↑↓)
Hund Half-fill all orbitals in a subshell with ↑ before any pairing
Every pair of neighboring atoms feels both a short-range repulsion (electron clouds resist overlap — Pauli exclusion) and a longer-range attraction (Coulombic / van der Waals / shared electrons).
At a unique distance r₀ these forces cancel exactly. That balance point is the equilibrium bond length — and the energy reaches its minimum, −Ebond.
The r−12 term dominates at short range → steep repulsive wall. The r−6 term dominates at long range → gentle attractive tail. Minimum at r₀ = 21/6 σ ≈ 1.122 σ.
The slope of the energy curve is the force. Where the curve is flat (r = r₀) the force is zero. Where the curve dives steeply, the force is large.
The repulsive wall (r < r₀) is much steeper than the attractive tail (r > r₀). Push the atoms closer → huge restoring force, they snap back. Pull them apart → only a mild pull, they drift back slowly (no sudden bounce). This is why solids resist compression strongly but stretch more readily — and why they ultimately break in tension once r exceeds the inflection point.
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