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# chem5314_hwk3 - Chem 5314 homework#3 due Oct 13 2011...

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Chem 5314 homework #3 – due Oct. 13, 2011 Problem 1 – harmonic oscillator wavefunctions In class, we found that the stationary states of the 1d harmonic oscillator have the form ψ n = A n ( x n +? x n - 2 +? x n - 4 + · · · ) e - αx 2 (1) where A n is a normalization constant, the polynomial in brackets ends with x or a constant if n is respectively odd or even, and where α = 2 ~ (2) We did not derive a general formula for the coefficients ( i.e. the ?’s) in the polynomial. These could , although its not very practical, be determined by orthogonality . For all of question 1, express all your answers and do all your work in terms of the parameter α only. a) In particular, the second excited state ψ 2 has the form ψ 2 = A 2 ( x 2 + c ) e - αx 2 (3) Find the constant c by requiring that ψ 2 be orthogonal to the ground state ψ 0 . b) ψ 2 is also orthogonal to the first excited state ψ 1 . Why? (hint: symmetry) c) Determine the normalization constant A 0 for the ground state.

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