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Ψ(x,0) = 1/sqrt(2)*Φ1 + sqrt(2/5)*Φ3 + 1/sqrt(10)*Φ5

where φn = sqrt(2/a)*sin(n*pi*x/a)

Is this unique why or why not? I'm thinking that it has something to do with all odd Energies.

Also is

Φ(x) = sin^2 (7*pi*x/a)

an acceptable state. To normalize this function would mean to integrate a sine with a power of 4, how can I do this. Also how can I express this state as a linear combination (mixed states) of energy eigenstates with appropriate coefficients?