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Eigenfunction Decomposition of 1DHO Wavefunctions

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A particle of mass m is subject to the one-dimensional harmonic oscillator potential. Write down the first three normalised eigenfunctions ?_n (x) and the corresponding eigenvalues.
Initially the wavefunction is in a mixed state of the form
?(x)=(1/(7???))^(1?2) e^(-x^2/(2?)^2 ) ((3x)^2/(?)^2 +(x/?)-(3/2)+?2)
where ?=?(??m?). Let ?(x) be written in terms of the normalised eigenfunctions of the harmonic oscillator
?(x)=?_(n=0)^?(c_n ?_n (x)).
Calculate the coefficients_n. Hence determine the possible outcomes E_n and associated probabilities of a measurement of the particle's energy. What will the energy be after making a measurement?

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This solution writes down the first three eigenfunctions of a one-dimensional harmonic oscillator and represent a given wavefunction as a linear combination of these eigenfunctions.

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