3.2 Hermite polynomials (weighting function not 1!) The time-independent ID Schr

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3.2 Hermite polynomials (weighting function not 1!) The time-independent ID Schrödinger equation for the harmonic oscillator (see p.51 of Griffiths 2nd edition) is Making the replacement that of the form ()-h(E)e-e/2, the Schrödinger equation reduces to 2m dr22 = zVfnu h and K-M/(h) and looking for solutions dhdh This can be rewritten in our familiar SL form as the (Physicist's) Hermite equation dh (a) Show how Equation (1) leads to Equation (2). And that Equation (3) reduces to (b) What is the orthogonality condition for eigenfunctions hn()? You should not need (c) Find the first four Hermite polynomials using a power series expansion. Each poly- Equation (2). to do a calculation here. nomial can be multiplied by any constant and remains an eigenfunction so don't worry about these constants. By doing this calculation, you also uncover what the allowed energies must be - what are they? Does this make sense? Aside: Usually the states of the harmonic oscillator have numbering starting at O Why might this be?

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https://drive.google.com/open?id=120WW-LTbSgfyhUdwyXryj5l7TLXahfEP

this is my answer as I am unable to type the eqations in the prescribed format.

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