The intensities of spectroscopic transitions between the vibrational states of a molecule are proportional to the square of the integral ∫ψv′xψvdx over all space. Use the relations between Hermite polynomials given in Table 7E.1 to show that the only permitted transitions are those for which v′ = v ± 1 and evaluate the integral in these cases. Table 7E.1 The Hermite polynomialsH,(y)112y24y – 238y' – 12y416у' — 48у + 1232y – 160y' + 120y64y - 480у + 720у- 120The Hermite polynomials are solutions of the differential equationH" – 2yH; + 2vH,=0where primes denote differentiation. They satisfy the recursion relationH,- 2yH, + 2vH, =0An important integral isif v'+vSH,H,e* dy=-TU 2" v! if v'=v

Answered: Table 7E.1 The Hermite polynomials…

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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The intensities of spectroscopic transitions between the vibrational states of a molecule are proportional to the square of the integral ∫ψ_v′xψ_vdx over all space. Use the relations between Hermite polynomials given in Table 7E.1 to show that the only permitted transitions are those for which v′ = v ± 1 and evaluate the integral in these cases.

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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