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
University Physics Volume 3
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Chapter7: Quantum Mechanics
Section: Chapter Questions
Problem 42P: Show that when 1(x,t) and 2(x,t) are solutions to the time-dependent Schrödinger equation and A, 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 ∫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.
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