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2. Hermite Polynomials The generating function of the Hermite polynomials is given by ∞ 2-s²+2sy = Σ H₂(y): n! sn n=0 a) Use this to derive analytic forms for Ho(y), H₁(y), and H₂(y). b) Proof that •+∞ е [ dy e - a² + 2ªve-l²+2tv e-v² = √ire ²01 c) Use Eqs. (1) and (2) to derive the orthogonality relation •+∞ dy Hn(y) Hm(y)e¯y² = √√2¹n!dnm.

2. Hermite Polynomials The generating function of the Hermite polynomials is given by ∞ 2-s²+2sy = Σ H₂(y): n! sn n=0 a) Use this to derive analytic forms for Ho(y), H₁(y), and H₂(y). b) Proof that •+∞ е [ dy e - a² + 2ªve-l²+2tv e-v² = √ire ²01 c) Use Eqs. (1) and (2) to derive the orthogonality relation •+∞ dy Hn(y) Hm(y)e¯y² = √√2¹n!dnm.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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2. Hermite Polynomials The generating function of the Hermite polynomials is given by +∞ a) Use this to derive analytic forms for Ho(y), H₁(y), and H₂(y). b) Proof that is •+∞ dy 8 ²+2sy e - Σ H₂ (y) 77. = n! n=0 +2sy _t²+2ty e -y² c) Use Eqs. (1) and (2) to derive the orthogonality relation = √πe²st dy Hn(y)Hm(y)e¯y² = √√π2ªn!dnm. with d) In class we have shown that the eigenfunctions of the harmonic oscillator are given by un (y) = CnH₂(y)e-²1 y ² Use Eq. (3) to determine the constant Cn therein. y = Μω ħ -X.
Transcribed Image Text:2. Hermite Polynomials The generating function of the Hermite polynomials is given by +∞ a) Use this to derive analytic forms for Ho(y), H₁(y), and H₂(y). b) Proof that is •+∞ dy 8 ²+2sy e - Σ H₂ (y) 77. = n! n=0 +2sy _t²+2ty e -y² c) Use Eqs. (1) and (2) to derive the orthogonality relation = √πe²st dy Hn(y)Hm(y)e¯y² = √√π2ªn!dnm. with d) In class we have shown that the eigenfunctions of the harmonic oscillator are given by un (y) = CnH₂(y)e-²1 y ² Use Eq. (3) to determine the constant Cn therein. y = Μω ħ -X.
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