The Quantum Simple Harmonic Oscillator
Verify the evenly-spaced energy ladder Eₙ = (n+½)ℏω, measure how spacing depends on mass and stiffness, and observe time-evolved superpositions.
A. Energy levels and wavefunctions
| n | Eₙ (eV) | Eₙ₊₁−Eₙ (eV) | nodes |
|---|---|---|---|
| 0 | 0.17992 | 0.35984 | 0 |
| 1 | 0.53977 | 0.35984 | 1 |
| 2 | 0.89961 | 0.35984 | 2 |
| 3 | 1.25945 | 0.35984 | 3 |
| 4 | 1.61930 | 0.35984 | 4 |
| 5 | 1.97914 | 0.35984 | 5 |
Eₙ = (n+½)ħω, ω = √(k/m). Level spacing ħω is constant — contrast with E ∝ n² for the square well. Click a level to select it.
B. Time development
Single eigenstate: |ψ|² is frozen. With n+1 mixed: probability density oscillates at beat frequency ω = (E₀₊₁ − Eₙ)/ħ, which here equals the classical angular frequency — because the ladder is evenly spaced.
Observations
Record your measured data in the tables below. Refer to the lab manual for the full procedure.
Table 6.1 — Energy levels and nodes (proton, default stiffness)
| n | Eₙ (eV) | Eₙ−Eₙ₋₁ (eV) | Nodes |
|---|---|---|---|
Table 6.2 — Level spacing E₁−E₀ for various settings
| Setting | E₁−E₀ (eV) |
|---|---|
Rows: Increased k; Electron (default k); Proton (default k); Heavy atom (default k)