Table 7E.1 The Hermite polynomials H,(y) 1 1 2y 2 4y – 2 3 8y' – 12y 4 16у' — 48у + 12 32y – 160y' + 120y 64y - 480у + 720у- 120 The Hermite polynomials are solutions of the differential equation H" – 2yH; + 2vH,=0 where primes denote differentiation. They satisfy the recursion relation H,- 2yH, + 2vH, =0 An important integral is if v'+v SH,H,e* dy=- TU 2" v! if v'=v
Table 7E.1 The Hermite polynomials H,(y) 1 1 2y 2 4y – 2 3 8y' – 12y 4 16у' — 48у + 12 32y – 160y' + 120y 64y - 480у + 720у- 120 The Hermite polynomials are solutions of the differential equation H" – 2yH; + 2vH,=0 where primes denote differentiation. They satisfy the recursion relation H,- 2yH, + 2vH, =0 An important integral is if v'+v SH,H,e* dy=- TU 2" v! if v'=v
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Calculate the values of ⟨x3⟩v and ⟨x4⟩v for a harmonic oscillator by using the properties of the Hermite polynomials given in Table 7E.1; follow the approach used in the text.
Transcribed Image Text:Table 7E.1 The Hermite polynomials
H,(y)
1
1
2y
2
4y – 2
3
8y' – 12y
4
16у' — 48у + 12
32y – 160y' + 120y
64y - 480у + 720у- 120
The Hermite polynomials are solutions of the differential equation
H" – 2yH; + 2vH,=0
where primes denote differentiation. They satisfy the recursion relation
H,- 2yH, + 2vH, =0
An important integral is
if v'+v
SH,H,e* dy=-
TU 2" v! if v'=v
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