Quantum Well Eigenstates (Box & HO)
This page plots stationary eigenfunctions ψ_n(x) and probability densities |ψ_n|² for two textbook potentials: an infinite square well on [0, L] and the quantum harmonic oscillator. Units are pedagogical (ħ = m = L = 1 for the well; ħ = m = ω = 1 for the oscillator). You can overlay a time-dependent phase factor e^{−iE_n t/ħ} to visualize how the complex amplitude rotates while |ψ|² remains fixed for a pure energy eigenstate.
Who it's for: Introductory quantum mechanics and modern physics courses comparing quantization in a box vs a parabolic well.
Key terms
- Infinite Square Well
- Harmonic Oscillator
- Hermite Polynomials
- Energy Quantization
- Probability Density
How it works
Pick a confining potential — infinite square well or harmonic oscillator — and the quantum number n. The canvas shows ψ_n (yellow) and probability density |ψ_n|² (blue). Toggle time phase to see stationary-state oscillation of the complex amplitude.
Frequently asked questions
- Why does turning on “phase animation” not change the blue |ψ|² bars for a single stationary state?
- For an eigenstate, the only time dependence is an overall phase e^{−iE_n t/ħ}. The probability density |e^{−iE_n t/ħ} ψ_n(x)|² = |ψ_n(x)|² is unchanged. The yellow trace shows the real part of the complex amplitude, which oscillates in time.
- How do the energy scales differ between the box and the oscillator?
- In the chosen units, the box has E_n ∝ n² (kinetic confinement on a segment), while the oscillator has E_n = (n + ½)ħω with ħω = 1 here. The shapes of ψ_n also differ: sines inside hard walls vs Hermite polynomials modulated by a Gaussian envelope.
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