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The task is to form 2 linear combinations out of the 2 given spherical harmonic functions. I dont understand why the resultant wave function has to be multiplied with the constant $1/sqrt(2)$?

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Without this factor the wavefunction would not be properly normalized. Normalization is necessary to make the integral over the probability amplitude be 1.

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This simply comes back to the fact these functions are just elements of some vector space. In this special case, the spherical harmonics are both normalized and orthogonal. By looking at them as simply vectors in a vector space then, it is easy to show that their sum would have a norm of $\sqrt{2}$. It seems you wish to normalize the result so you divide by that factor.

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