Hermite Polynomial Calculator
Free online hermite polynomial calculator. Enter your values and get instant results with the formula, step-by-step working, a worked example and FAQs — everything runs locally in your browser on Calcton, no signup required.
中文版：埃尔米特多项式在线计算器
埃尔米特多项式 Hn（物理学家约定）由三项递推 Hn+1 = 2x·Hn − 2n·Hn−1 生成，从 H0 = 1、H1 = 2x 起步。它带着权重 e^(−x²) 在整条实轴上正交：∫ Hm·Hn·e^(−x²) dx = √π·2ⁿ·n!·δ(m,n)。
它是量子谐振子的语言：第 n 个定态波函数正比于 Hn(√2·x/ℓ)·e^(−x²/2ℓ²)，能级 (n + 1/2)ℏω。概率论里它同样出场——高斯分布的 n 阶矩与 Hermite 展开一一对应（高斯求积的节点正是 Hn 的根）。
参考值
H5(1) = −8，H6(1) = 184，H7(1) = 464
H5(0.5) = 41，H4(2) = 76
正交性：∫ H2·H3·e^(−x²) dx = 0，∫ H2²·e^(−x²) dx = 8√π = 14.17963080724413
How to use the Hermite Polynomial Calculator
- Fill in the inputs with your own numbers — every field shows what it expects.
- The result updates instantly as you type; no "Calculate" click needed.
- Read the result card for the main figure, the step-by-step formula, and practical notes.
Frequently asked questions
Is the Hermite Polynomial Calculator free?
Yes — all math calculators on Calcton are free, with no sign-up required. Your inputs are processed in your browser and never sent to a server.
Is there a Chinese version of this tool?
Yes. Use the Chinese version link above — it is the same calculator with the interface in Simplified Chinese.
References
Formulas and benchmarks used by this calculator follow these published sources.
KevinContent LeadCalcton Editorial Team · Last updated 2026-09-28
Kevin maintains Calcton's formula verification and content: every calculator cites the standard it implements, worked examples are recomputed programmatically, and health benchmarks follow WHO and international guidelines.
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