The quantum harmonic oscillator

Equipment

• 3D canvas plotting probability density and energy levels, rotated by dragging or arrow keys
• Slider «Số lượng tử n» selecting the eigenstate from n = 0 to n = 5
• Readout line showing «Eₙ/ℏω» and the spacing «Eₙ₊₁ − Eₙ = ℏω»

Procedure

1. Observe the ground state n = 0

   Set «Số lượng tử n» = 0: the canvas shows a bell-shaped probability density with no node, and the readout reports «Eₙ/ℏω = 0.5». Note the zero-point energy $E_0 = \tfrac{1}{2}\hbar\omega \neq 0$ — the particle cannot rest in the parabolic well.

2. Sweep the excited levels and check the even spacing

   Step «Số lượng tử n» through 0 → 1 → 2 → 3 → 4 → 5. Each time record «Eₙ/ℏω» and check that consecutive levels differ by «Eₙ₊₁ − Eₙ = ℏω». Compare with $E_n = \hbar\omega\left(n + \tfrac{1}{2}\right)$: the readings must be 0.5, 1.5, 2.5, 3.5, 4.5, 5.5 in turn.

3. Count nodes and study the probability density shape

   At each level n, drag the canvas to rotate the view and count the nodes (zero-density points) of the probability density: exactly n of them, with tails leaking into the classically forbidden region. Observe that for large n the density concentrates near the two turning points, approaching the classical distribution.

4. Compare with the classical oscillator and predict

   Before moving «Số lượng tử n» to a new value, predict «Eₙ/ℏω» using $E_n = \hbar\omega\left(n + \tfrac{1}{2}\right)$ and check the readout. Conclude: unlike the classical oscillator that accepts any continuous energy, the quantum oscillator admits only evenly spaced levels with lowest energy $\hbar\omega/2$.

Simulation

Experiment history

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