Quantum Mechanical Model of Atom
Unit 2 moves from Bohr's fixed orbits to orbitals described by wave functions, and examines the quantum numbers and filling rules that generate the periodic table.
Core rules
- de Broglie relation: λ = h/mv = h/p, so a particle's wavelength falls as mass and speed increase.
- Heisenberg uncertainty principle: Δx·Δp ≥ h/4π; position and momentum cannot both be fixed.
- ψ has no direct physical meaning, but ψ² gives probability density; an orbital is a region of high probability.
- Quantum numbers: n (size and energy), l = 0 to n−1 (s, p, d, f), m = −l to +l, and s = ±½.
- Shell n contains n² orbitals and holds 2n² electrons; a d subshell has 5 orbitals.
- Nodes: radial = n − l − 1, angular = l, total = n − 1.
- Electrons fill by the (n + l) rule, then lower n; Pauli excludes identical quantum-number sets, and Hund keeps parallel spins in degenerate orbitals.
Worked example
Find the de Broglie wavelength of an electron accelerated through 100 V. Using λ = h/√(2meV) = 1.227/√V nm, λ = 1.227/10 nm = 0.1227 nm ≈ 1.23 × 10⁻¹⁰ m. Substituting h = 6.626 × 10⁻³⁴ J s, m = 9.11 × 10⁻³¹ kg, e = 1.602 × 10⁻¹⁹ C and V = 100 V gives the same value.
Traps
- Spin is not obtained from the Schrödinger equation; it is an additional quantum number.
- 4s fills before 3d, but once filled, 3d electrons are lost first in transition metals.
- l cannot equal n; for n = 2 only 2s and 2p exist.
- Do not confuse an orbit (fixed path) with an orbital (probability region).