Hermite Polynomials: In studying the quantum harmonic oscillator, one ends up with the following differential equation: y" – 2xy' + 2ky = 0 ke R a) Show that if y(x) n = 0, 1, ... 2(n-k) (n+2)(n+1)On for E-o Onx" solves the differential equation, then ơn+2 b) Show that for k e N, there exists a polynomial solution. For k = 3, exhibit the unique monic polynomial solution.

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Hermite Polynomials: In studying the quantum harmonic oscillator, one ends up with the following
differential equation:
y" – 2xy' + 2ky = 0
ke R
a) Show that if y(x)
n = 0, 1, ...
2(n-k)
(n+2)(n+1)On for
E-o Onx" solves the differential equation, then ơn+2
b) Show that for k e N, there exists a polynomial solution. For k = 3, exhibit the unique monic
polynomial solution.
Transcribed Image Text:Hermite Polynomials: In studying the quantum harmonic oscillator, one ends up with the following differential equation: y" – 2xy' + 2ky = 0 ke R a) Show that if y(x) n = 0, 1, ... 2(n-k) (n+2)(n+1)On for E-o Onx" solves the differential equation, then ơn+2 b) Show that for k e N, there exists a polynomial solution. For k = 3, exhibit the unique monic polynomial solution.
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