Question: The Hermite polynomials are defined by (a) Verify that H0 = 1, H1 = 2z, H2 = 4z2 - 2, H3 = 8z3 - 12z

The Hermite polynomials are defined by

(a) Verify that
H0 = 1, H1 = 2z, H2 = 4z2 - 2, H3 = 8z3 - 12z
(b) The Hermite polynomials obey the relation (Pauling and Wilson, pages 77-79)
zHn(z) = nHn-1(z) + 1/2 Hn+1(z)
Verify this identity for n = 0, 1, and 2. (c) The normalized harmonic-oscillator wave functions can be written as (Pauling and Wilson, pages 79-80) cv1x2 = 12v
Ïˆv (x) = (2v v!)-1/2 (Î±/Ï€)1/4 e-Î±x2/2 Hv(Î±1/2x)
Verify (4.86) for the three lowest states.

H,(c) = (-1 rear de

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