What is the ultraviolet catastrophe?
Classical physics prediction that blackbody radiators emit infinite energy at high frequencies. Rayleigh-Jeans: u(v,T) = 8πv²kT/c³ → diverges. Fixed by Planck’s quantization. MNEMONIC: ‘UV = Ultraviolet Violence — classical physics self-destructs’
Planck’s quantum hypothesis
Energy comes in discrete packets: E = hν = ℏω. Planck’s constant h = 6.626×10⁻³⁴ J·s. This was the birth of quantum mechanics. MNEMONIC: ‘Energy Has Frequency’ — E=hν
Photoelectric effect equation
KE_max = hν − Φ. Photon energy minus work function. Only frequency matters, not intensity. Einstein won Nobel for this. MNEMONIC: ‘Kick Energy = Hit minus Fight (threshold)’
Bohr’s hydrogen energy levels
E_n = −13.6 eV/n². Ground state n=1 is −13.6 eV. MNEMONIC: ‘Thirteen-six over n-squared — negative because bound’
de Broglie wavelength
λ = h/p = h/mv. Particles have wave properties. MNEMONIC: ‘Lambda = H/P — HiP wavelength’
Wave-particle duality
All matter has both wave and particle properties. Demonstrated by electron diffraction (Davisson-Germer 1927). Interference pattern in double-slit destroyed by which-path observation. MNEMONIC: ‘Wave-Particle: Both or Neither, Never Just One’
Group velocity vs phase velocity
Group velocity = dω/dk = p/m = classical velocity. Phase velocity = ω/k ≠ physical velocity for massive particles. MNEMONIC: ‘Group Goes with the particle. Phase is just a Phase.’
Born probability interpretation
|ψ(x,t)|² dx = probability of finding particle in dx. Normalization: ∫|ψ|²dx = 1. MNEMONIC: ‘Born Rule: Modulus-Squared is the Messenger’
Time-dependent Schrödinger equation
iℏ ∂ψ/∂t = Ĥψ = [−ℏ²/2m ∂²/∂x² + V]ψ. Deterministic equation of motion for wavefunction. MNEMONIC: ‘I Have a Hamiltonian’ — iℏ = Ĥ
Time-independent Schrödinger equation
Ĥφ = Eφ. Eigenvalue equation. Applies when V is time-independent. Solutions are stationary states. MNEMONIC: ‘H-hat equals E: the Hamiltonian Eigenvalue Equation’
Expectation value
<a> = ∫ψ* Â ψ dx = <ψ|Â|ψ>. Predicted average of many measurements on identical states. MNEMONIC: ‘Sandwich the operator between bra and ket’</a>
Infinite square well energy levels
E_n = n²π²ℏ²/(2mL²) = n²h²/(8mL²). Scales as n². MNEMONIC: ‘n-squared ladder in the box — energy climbs as square of floor number’
Infinite square well wavefunctions
φ_n(x) = √(2/L) sin(nπx/L). n nodes corresponds to n antinodes. MNEMONIC: ‘Root-2-over-L times sine of n-pi-x-over-L’
Zero-point energy
E_1 = π²ℏ²/(2mL²) > 0. Particle cannot be at rest in a box — required by Heisenberg. MNEMONIC: ‘Even at zero temperature, quantum particles NEVER rest’
Superposition in a well
ψ = Σ c_n φ_n. |c_n|² = probability of measuring E_n. After measurement, state collapses to φ_n. MNEMONIC: ‘Coefficients Squared = Chances’
Heisenberg Uncertainty Principle
σ_x σ_p ≥ ℏ/2. Fundamental property of waves, NOT a measurement disturbance artifact. MNEMONIC: ‘X times P is at least Half-ℏ — you can’t have both sharp’
Energy-time uncertainty
ΔE·Δt ≥ ℏ/2. Excited states have energy width ΔE ~ ℏ/2τ where τ is lifetime. Basis for natural linewidth. MNEMONIC: ‘Sharp energy needs eternal time — short life means fuzzy energy’
Canonical commutator
[x̂, p̂] = iℏ. Source of all position-momentum uncertainty. Non-commuting operators are incompatible observables. MNEMONIC: ‘x-p commutator = iℏ — the heart of quantum mechanics’
Minimum uncertainty state
Gaussian wavepacket achieves σ_x σ_p = ℏ/2 exactly. All others exceed this. Harmonic oscillator ground state is minimum uncertainty. MNEMONIC: ‘Gaussians are quantum perfection — minimum uncertainty’
Harmonic oscillator energy levels
E_n = (n + ½)ℏω, n = 0,1,2,… Equally spaced by ℏω. Zero-point energy ℏω/2. MNEMONIC: ‘n-plus-a-half ℏω — never zero, always half’
Ladder operators
â = √(mω/2ℏ)(x̂ + ip̂/mω), ↠= raising. [â, â†] = 1. â†|n⟩ = √(n+1)|n+1⟩, â|n⟩ = √n|n−1⟩, â|0⟩ = 0. MNEMONIC: ‘Raise with dagger, lower without — can’t go below ground’
Coherent states
Eigenstates of lowering operator: â|α⟩ = α|α⟩. Minimum uncertainty at all times. Most classical-like quantum state. Basis of laser physics. MNEMONIC: ‘Coherent = Classical-looking quantum — lasers run on these’
Why harmonic oscillator is universal
Any smooth potential looks quadratic near minimum (Taylor expansion). All molecules, phonons, photon modes are harmonic oscillators. QFT = infinite collection of QHOs. MNEMONIC: ‘Everything wobbles like a spring near equilibrium’
Hermitian operators
 = †. Real eigenvalues. Orthogonal eigenstates. All physical observables must be Hermitian. MNEMONIC: ‘Hermitian = Honest — gives REAL answers’
