Hermite Polynomial | Santa Fe College - Edubirdie

University:
Santa Fe College
Course:
PHY 2048 | General Physics 1 with Calculus
Academic year:

2024
Views:

410

Pages:

3
Author:

Jabin F.

Hermite polynomial [H_{n}(y)=(-1)^{n}e^{y^{2}}(\frac{q}{dy})^{n}e^{-y^{2}}] H0(y) = 1 H1(y) = 2y H2(y) = 4y^2 - 2 ψ0(x) = (α / 4√π)^(1/2) * e^(-α^2 * x^2 / 2) ψ1(x) = (α / π)^(1/2) * e^(-α^2 * x^2 / 2) * (2αx) ψ2(x) = (α / 4 * 2√π)^(1/2) * e^(-α^2 * x^2 / 2) * (4α^2 * x^2 - 1) For Symmetric V(-n) = V(n) ψn(x) have definite parity ψn(-x) = +ψn(x) = even, n = 0, 2, 4, 6, ... ψn(-x) = -ψn(x) = odd, n = 1, 3, 5, 7, ... The no. of nodes = n ψₙ(x) = {even, n = even, 2, 4, 6; odd, n = odd, 1, 3, 5, ...} ψₙ(x) = {zero at x = 0, n = odd; Non-zero at x = 0, n = even}

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