Hermite polynomials: H_n(x) orthogonal with weight e^{-x²}. Physicist: H_0=1, H_1=2x, H_2=4x²−2. Probabilist: He_n(x)=2^{-n/2}H_n(x/√2). Quantum harmonic oscillator: ψ_n(x) = (1/√(2^n·n!·√π))H_n(x)e^{-x²/2}.
How to Use
Enter n and x:
- H_n(x): Physicist convention
- He_n(x): Probabilist
- Sequence: Step values
Quantum Mechanics
The quantum harmonic oscillator has energy levels E_n=(n+½)ℏω. The wavefunctions are ψ_n(x)∝H_n(αx)exp(-α²x²/2) where α=√(mω/ℏ). The n-th excited state has n nodes (zeros).
Two Conventions
Physicist: H_n, weight e^{-x²}, H_n(0)=(-1)^{n/2}·n!/(n/2)! for even n. Probabilist: He_n, weight e^{-x²/2}, monic. Relation: H_n(x)=2^{n/2}He_n(x√2). Always check which convention!
Step-by-Step Instructions
- 1Enter degree n.
- 2Enter x value.
- 3Compute H_n(x).
- 4See probabilist.
- 5View QHO wavefunction.