7Experiment 7
Free-Particle Wave Packets and Unbound States
Watch an exact Gaussian wave packet propagate and spread. Measure group velocity vs. phase velocity and how the spreading time τ depends on width and mass.
A. Wave packet parameters
1.00 nm
0.30 eV
k₀ = √(2mEₖ)/ħ2.806e+9 m⁻¹
v_g = ħk₀/m324.86 km/s
v_p = ħk₀/2m = v_g/2162.43 km/s
τ = 2md²/ħ17.27 fs
Δx·Δp (t=0)0.5000 ħ = ħ/2 ✓
A Gaussian wave packet saturates the Heisenberg uncertainty relation Δx·Δp = ħ/2 at t=0. Spreading increases Δx while Δp stays constant.
σ(t) spreading curve
σ(t) = d√[1+(t/τ)²] — exact formula; red dot = current time.
1.0
Re ψ(x,t) |ψ(x,t)|² ±σ(t) envelope
⟨x⟩(t) = 0.000 nm
σ(t) = 0.000 nm
(exact from closed-form solution — not from numerical integration)
Observations
Record your measured data in the tables below. Refer to the lab manual for the full procedure.
Table 7.1 — Spreading time: width and mass dependence
| Setting | d (nm) | Particle | τ (fs) |
|---|---|---|---|
Rows: Baseline (d=1.0 nm, e⁻); Narrower (d=0.3 nm, e⁻); Proton (d=1.0 nm)
Table 7.2 — Group and phase velocities at two kinetic energies (electron)
| KE (eV) | v_g (km/s) | v_p (km/s) | |
|---|---|---|---|