# Combine the completeness and linearity of Schrodinger equation's solution, can we say that any wavefunction can solve any Schrodinger equation?

From quantum mechanics, I learnt:

1. Completeness of energy eigenfunction: The energy eigenfunctions of the Schrodinger equation span the space, i.e. any state can be expanded as linear combination of energy eigenstates.\ That is $$\Psi(x)=\sum_{n=0}^{\infty}b_{n}\psi_{n}(x)$$.
And $$\Psi(x,t)=\sum_{n=0}^{\infty}b_{n}e^{-\frac{itE_{n}}{\hbar}}\psi_{n}(x)$$

2. Linearity of Schrodinger equation: Linear combination of solutions of Schrodinger equation can also solve the Schrodinger equation.

Therefore, can we say that any wavefunction can solve an arbitrary Schrodinger equation? But that sounds weird.

The question actually comes from the MIT8.06 lecture (1:02:46), where the professor stated that the trial wave function is not the solution of Schrodinger equation, but can be expanded as energy eigenstates, so I'm confused.

• For the last paragraph, is it mean that $\Psi(x,t)=\sum_{n=0}^{\infty}b_{n}e^{-\frac{itE_{n}}{\hbar}}\psi_{n}(x)$ is not a solution of time dependent SE, becuase it is not a linear superposition? Jul 23, 2022 at 8:14