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Wave Packet & Dispersion
Superpose cos(kx−ωt); ω=ck+αk²; spreading vs Wave on String PDE.
Group & Phase Velocity
Two-wave beat: ω(k)=ck+αk²; v_g=Δω/Δk vs v_p=ω̄/k̄; envelope vs carrier.
Coherent State |α⟩ in a Harmonic Oscillator
Animated Gaussian wavefunction of a coherent state |α⟩ in a 1-D harmonic well: rigid σ = 1/√2 packet whose centroid traces the classical orbit ⟨x⟩(t) = √2|α|cos(ωt − φ_α). Side-by-side phase space, |ψ(x,t)|², and ⟨x⟩(t) trace.
Phonon Dispersion: 1-D Mass–Spring Chain
Monoatomic ω = 2√(K/m)|sin(ka/2)| or diatomic acoustic/optical branches from alternating masses; ω(k) and group velocity v_g = dω/dk in the first Brillouin zone with animated lattice snapshot.
Particle in a 1D Box
Superposition of n=1 and n=2: time evolution of |Ψ|² and ⟨E⟩ (model units).
Heisenberg Product σ_x σ_p
Gaussian position and momentum widths with ℏ = 1; momentum “excess” over the minimum-uncertainty floor.