The Gays-Hermite quadrature is defined for integrals of the form: I= 1 = [__ 1(x) e...

 

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The Gays-Hermite quadrature is defined for integrals of the form: I= 1 = [__ 1(x) e ==² which has the following representation in quadratures: N da I= Σw.f(x₂) i=1 Rodrigues: where the points xi are the roots of the Hermite polynomials given by the formula d den H₂(E)= (-1)"e² The formula for the Gaus-Hermite weights is given by: 2n-¹n! √ n²H₁-1(x)2 a) Find the first 20 zeros of the polynomials and the corresponding weights of the quadrature. (b) The state of a harmonic oscillator in quantum mechanics is given by the functions probability: 1 mw ¹/4e-²/2 H₂(E) 2n! Th = where {=pmw-x. Make pmw~= 1, that is, & = x for the numerical application. Numerically estimate the root mean square value of the position of the particle in the first exited state (n = 1). The exact value of the integral is given by < 2²³ >= * | ₁(a)³²2²dx = 3/2 The Hermite polynomial of order 1 is given by: H1(x) = 2x The Gays-Hermite quadrature is defined for integrals of the form: I= 1 = [__ 1(x) e ==² which has the following representation in quadratures: N da I= Σw.f(x₂) i=1 Rodrigues: where the points xi are the roots of the Hermite polynomials given by the formula d den H₂(E)= (-1)"e² The formula for the Gaus-Hermite weights is given by: 2n-¹n! √ n²H₁-1(x)2 a) Find the first 20 zeros of the polynomials and the corresponding weights of the quadrature. (b) The state of a harmonic oscillator in quantum mechanics is given by the functions probability: 1 mw ¹/4e-²/2 H₂(E) 2n! Th = where {=pmw-x. Make pmw~= 1, that is, & = x for the numerical application. Numerically estimate the root mean square value of the position of the particle in the first exited state (n = 1). The exact value of the integral is given by < 2²³ >= * | ₁(a)³²2²dx = 3/2 The Hermite polynomial of order 1 is given by: H1(x) = 2x

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The first 20 zeros of the Hermite polynomials and their corresponding weights for the GaussHermite q... View the full answer