# 量子谐振子 - Quantum Harmonic Oscillator

量子谐振子能级和波函数的交互式可视化

> 标准页面: https://elysiatools.com/zh/visualizations/quantum-harmonic-oscillator

- **分类:** Physics

## 概述

Interactive visualization of quantum harmonic oscillator demonstrating equally spaced energy levels, parabolic potential well, and Hermite polynomial wave functions. Features the fundamental quantum system V(x) = ½mω²x² with characteristic properties: zero-point energy E₀ = ½ħω (ground state energy due to uncertainty principle), equally spaced energy levels Eₙ = (n + ½)ħω where adjacent levels are separated by exactly ħω, wave functions ψₙ(ξ) = Nₙ·Hₙ(ξ)·e^(-ξ²/2) using Hermite polynomials Hₙ(ξ) with dimensionless coordinate ξ = √(mω/ħ)·x. Each energy level has exactly n nodes (where ψ = 0) showing increasing complexity with quantum number. Parabolic potential visualization with classical turning points x_tp = ±√(2Eₙ/mω²) shown as dashed lines. Real-time wave function display with real part Re[ψ] (blue) and probability density |ψ|² (purple). Energy level ladder diagram showing first 15 levels with equal spacing characteristic, current level highlighting with energy value, and wave function thumbnails. Node visualization with labeled node positions (n₁, n₂, ...). Probability density display with classical region highlighting between turning points and quantum tunneling effect outside. Transition visualization showing absorption (n→n+1, upward green arrow) and emission (n→n-1, downward red arrow) processes with photon emission/absorption diagrams. Classical turning points visualization showing forbidden region penetration depth decreasing with energy. Expected position calculation ⟨x⟩ = 0 for all eigenstates (due to symmetry). Adjustable parameters: oscillator frequency ω (0.5-3.0 rad/fs), particle mass m (0.1-5.0 electron masses), quantum number n (0-10), maximum display levels (3-15). Quick presets: ground state (n=0), first excited (n=1), high energy (n=5), classical comparison (n=10). Display options: toggle real part, probability density, nodes, classical turning points. Transition options: initial level n_i, final level n_f, absorption/emission buttons. Educational content covers harmonic oscillator concept, parabolic potential and Hooke's law, Hermite polynomials (H₀=1, H₁=2ξ, H₂=4ξ²-2, H₃=8ξ³-12ξ, etc.), equally spaced levels (unique property leading to coherent states), zero-point energy and quantum fluctuations, classical correspondence (probability concentrates at turning points for large n), applications (molecular vibrations with vibrational spectra, phonons in solid state physics, quantum field theory foundation, coherent states in quantum optics, laser physics). Multi-language support (zh, en, es, fr, de, ru, pt).

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