How can I define what is the wavefunction

[Mancho]

How can I define what is the wavefunction if I'm given eigenvectors V1, V2,...Vn and energies E1, E2,. ..En.
I know that it must be a linear combination but how about constants?

 

[HallsofIvy]

I'm not expert in physics but in Linear Algebra, If we know the eigenvalues (energies here) and corresponding eigenvectors, we can construct the invertible matrix P, having those eigenvectors as columns, and the diagonal matrix D, having the corresponding eigenvalues on the diagonal then the matrix (operator) itself is given by
$$A= PDP^{-1}$$.

I'm sure there is something similar for quantum physics.

 

[dextercioby]

If the eigenvectors pertain to a compact selfadjoint operator, then any Hilbert space vector in which the operator acts can be chosen as a valid <wavefunction>. As for the uniqueness of the constants, well, it's very easy to prove, because the eigenvectors are perpendicular one to another and can be normalized to modulus 1, so that from (the sum in RHS converges weakly to the vector in the LHS)

$$\Psi = \sum_k a_k \psi_k$$

it follows that, for example,

$$a_3 = \langle \psi_3, \Psi\rangle$$