Module 3: Quantum Mechanics

Complete Notes with Formulas and Clear Physical Explanation

1. Black Body Radiation

A perfect black body absorbs all radiation and re-emits depending only on temperature.

Law	Formula	Remarks
Stefan-Boltzmann Law	P = σ A T⁴	Total power radiated ∝ T⁴ σ = 5.67 × 10⁻⁸ W/m²K⁴
Wien’s Displacement Law	λ_max T = 2.898 × 10⁻³ m·K	Peak wavelength decreases as T increases
Rayleigh-Jeans Law (classical)	u(λ) dλ = (8π kT / λ⁴) dλ	Works at long λ, fails at short λ → Ultraviolet catastrophe

2. Planck’s Quantum Hypothesis (1900)

Energy of oscillator of frequency ν is quantized:

E = n h ν (n = 0,1,2,...)

Planck’s Radiation Law
u(ν) dν = ^{8π h ν³}⁄_c³ ¹⁄_{e^hν/kT − 1} dν

u(λ) dλ = ^{8π h c}⁄_λ⁵ ¹⁄_{e^hc/λkT − 1} dλ

Perfectly matches experiment → Birth of Quantum Theory

3. Wave-Particle Duality

Phenomenon	Particle nature	Wave nature
Light	Photoelectric effect, Compton effect	Diffraction, Interference
Electrons	—	Davisson-Germer experiment (electron diffraction)

4. de Broglie Hypothesis (1924) – Matter Waves

λ = ^h⁄_p = ^h⁄_{m v} (h = Planck’s constant)

All particles have wave nature. Verified by electron diffraction.

5. Schrödinger Wave Equation

Time-dependent Schrödinger equation (general)

i ħ ^∂ψ(x,t)⁄_∂t = − ^ħ²⁄_2m ^∂²ψ⁄_∂x² + V(x) ψ = 0

Time-independent (stationary state) Schrödinger equation

− ^ħ²⁄_2m ^d²ψ⁄_dx² + V(x) ψ(x) = E ψ(x)

6. Born Interpretation of Wave Function

|ψ(x,t)|² dx = Probability of finding the particle between x and x+dx at time t

ψ itself has no direct physical meaning, only |ψ|² is measurable.
Wave function must be normalizable: ∫|ψ|² dx = 1

7. Particle in a One-Dimensional Infinite Potential Box (0 < x < L)

Potential: V(x) = 0 for 0 < x < L
V(x) = ∞ elsewhere → ψ = 0 outside the box

Solution of stationary Schrödinger equation:

ψ_n(x) = √²⁄_L sin(^{n π x}⁄_L) n = 1,2,3,...

Energy Eigenvalues (Quantized Energy)
E_n = ^{n² π² ħ²}⁄_{2 m L²} = ^{n² h²}⁄_{8 m L²}

Key results:

Energy is quantized (n = 1,2,3,...)
Zero-point energy: E₁ > 0 → cannot have zero energy (unlike classical)
Wave function has nodes
Probability density |ψ|² shows standing waves

8. Compton Effect (1923) – Proof of Photon Momentum

X-ray photon collides with free electron → wavelength increases.

Δλ = λ' − λ = ^h⁄_{m_e c} (1 − cos θ)

h/m_ec = 0.00243 nm = Compton wavelength

Proves light carries momentum p = h/λ

Summary Table of All Important Formulas

Concept	Formula	Meaning
Planck's quantum	E = h ν	Energy of photon
de Broglie wavelength	λ = h / p	Matter wave
Planck's law (frequency)	u(ν)dν = 8π h ν³/c³ × 1/(e^hν/kT−1) dν	Correct black body spectrum
Time-independent Schrödinger eq.	−(ħ²/2m) d²ψ/dx² + Vψ = Eψ	Stationary states
Particle in box energy	E_n = n² h² / (8 m L²)	Energy quantization
Particle in box wave function	ψ_n(x) = √(2/L) sin(nπx/L)	Standing wave
Compton scattering	Δλ = h/(m_ec) (1 − cosθ)	Photon momentum
Born interpretation	\|ψ\|² = probability density	Physical meaning of ψ

Quantum Mechanics begins where classical physics fails:
Black body radiation → Photoelectric effect → Compton effect → Wave nature of matter → Schrödinger equation
All experiments confirm these formulas with extreme precision.