Laguerre polynomials: L_n(x) orthogonal on [0,∞) with weight e^{-x}. L_0=1, L_1=1−x, L_2=(x²−4x+2)/2. Recurrence: (n+1)L_{n+1}=(2n+1−x)L_n−nL_{n-1}. Rodrigues: L_n=(e^x/n!)d^n/dx^n(x^ne^{-x}).
How to Use
Enter n and x:
- L_n(x): Evaluation
- Sequence: L_0..L_n
- Coefficients: Polynomial form
Hydrogen Atom
Radial wavefunction R_{nl}(r) ∝ L_{n-l-1}^{2l+1}(2r/na₀)·e^{-r/na₀}·(2r/na₀)^l. The associated Laguerre L_n^α determines the radial probability distribution. n−l−1 radial nodes.
Gauss-Laguerre
Zeros of L_n are nodes for Gauss-Laguerre quadrature: ∫_0^∞ e^{-x}f(x)dx ≈ Σw_if(x_i). Essential for integrals over [0,∞) with exponential decay. Exact for polynomials up to degree 2n-1.
Step-by-Step Instructions
- 1Enter degree n.
- 2Enter x value.
- 3Compute L_n(x).
- 4View sequence.
- 5See hydrogen.