Quantum Mechanics
Quantum Harmonic Oscillator
Explore the 1D Quantum Harmonic Oscillator $V(x) = \frac{1}{2}m\omega^2 x^2$. Visualize Hermite-Gaussian stationary states $\psi_n(x)$, discrete ladder energy levels $E_n = \hbar\omega(n + 1/2)$, and oscillating coherent state wave packets.
Quantum State n
n = 3
Energy Level E_n
3.50 ℏω
Expectation ⟨x⟩
0.000
Uncertainty Δx·Δp
0.500 ℏ
Quantum State Selector
Visual Overlay
Controls
Quantum Harmonic Oscillator Equations
Hamiltionian: Ĥ = - (ℏ² / 2m) d²/dx² + ½ m ω² x²
Eigenvalues: E_n = ℏω (n + ½) for n = 0, 1, 2, ...
Eigenstates: ψ_n(x) = (mω / πℏ)^(¼) · (1 / √(2ⁿ n!)) · H_n(ξ) · e^(-ξ²/2)
Eigenvalues: E_n = ℏω (n + ½) for n = 0, 1, 2, ...
Eigenstates: ψ_n(x) = (mω / πℏ)^(¼) · (1 / √(2ⁿ n!)) · H_n(ξ) · e^(-ξ²/2)
Developer Reference
Core Algorithm & Standalone Script
// Quantum harmonic oscillator Hermite polynomial solver
const canvas = document.getElementById('canvas');
const ctx = canvas.getContext('2d');
// E_n = hbar * omega * (n + 0.5)