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.

Quantum Well Eigenstates (Box & HO)

Stationary ψ_n and |ψ_n|² for infinite square well or harmonic oscillator; optional phase animation (ℏ = 1).

About this model

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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