Introduction to Quantum Mechanics

Introduction to Quantum Mechanics

1. Why Quantum Is Needed — Failures of Classical Physics

By 1900, classical physics explained planetary motion, waves, thermodynamics, but failed at microscopic / high-frequency:

• Blackbody radiation: Rayleigh-Jeans $u ∝ ν² kT$ predicted $u→∞$ at high $ν$ (ultraviolet catastrophe) vs observed peak and fall.
• Photoelectric effect: electrons ejected only if $hν > φ$ (work function), kinetic $K_{max}=hν-φ$, independent of intensity — classical wave predicted $K$ ∝ intensity and no threshold.
• Compton scattering: X-ray photon scatters with wavelength shift $Δλ = (h/mc)(1-cosθ)$ — particle-like momentum.
• Atomic spectra: discrete lines, not continuous; Rutherford atom unstable classically (orbiting electron should radiate and collapse).
• Specific heats at low $T$ drop, not constant $3R$.

All point to energy quantization.

2. Planck's Breakthrough (1900)

Planck's hypothesis: oscillators in cavity emit/absorb energy in quanta

with $h=6.626×10^{-34}$ J·s. Then average energy $⟨E⟩ = hν/(e^{hν/kT}-1)$ not $kT$, giving Planck's law:

which fits blackbody spectrum, reduces to Rayleigh-Jeans for $hν≪kT$ and Wien for $hν≫kT$.

3. Einstein's Steps (1905–1917)

• Photon: light itself is quanta

— explains photoelectric: $hν = φ + K_{max}$.

• A & B coefficients (1917) — stimulated emission already covered, quantized fields.

4. Bohr Model Prelude

Bohr quantized angular momentum $L=nħ$ ($ħ=h/2π$) for hydrogen to get discrete levels $E_n = -13.6 eV/n²$. Useful but ad hoc; true quantum mechanics replaces orbits with wavefunctions.

5. Core Ideas Ahead (preview next topics)

• De Broglie (1924): matter has wavelength $λ=h/p$.
• Heisenberg: $ΔxΔp≥ħ/2$, no simultaneous exact position/momentum.
• Schrödinger equation: wave equation for $ψ(x,t)$ whose $|ψ|²$ is probability.

6. Exam Points

• List failures of classical physics (blackbody, photoelectric, Compton, atomic spectra) — 5 marks, very frequent (almost certain to ask "why quantum needed").
• State Planck's $E=nhν$ and Planck's law — 5 marks, frequent.
• Photoelectric equation $hν=φ+K_{max}$ — 3-5 marks, frequent.
• Distinguish Rayleigh-Jeans vs Planck — 3 marks.