# On solutions of Schrödinger equation

Consider a quantum system described by the wave function $$\psi({\bf x}, t)$$ and subjected to a time-independent ordinary potential $$V({\bf x})$$. The relative Schrödinger equation takes the form: $$\underbrace{\left (-\frac {\hbar ^2 \nabla ^2}{2m} + V({\bf x}) \right)}_{\hat H ({\bf x}, \ {\bf p})} \psi({\bf x}, t) = i \hbar \frac \partial {\partial t}\psi({\bf x}, t)$$

Using separation of variables, we write the solution as $$\psi({\bf x}, t) = \varphi ({\bf x}) \ \phi(t)$$, where:

• $$\phi(t) = \exp(-\frac i \hbar Et)$$ solves the equation $$\displaystyle i \hbar \frac d {dt} \phi(t) = E \phi(t)$$ , with $$E$$ constant.
• $$\varphi({\bf x})$$ solves the equation $$H({\bf x},{\bf p}) \varphi ({\bf x})= E \varphi({\bf x})$$

Question. Is this the most general solution? Or is it just a particular one for a specific guess (a factorized solution)?

• Does this answer your question ? Jul 15, 2021 at 9:29

$$\psi(x,t) = \frac1{\sqrt2}\left(\phi_1(x) e^{-iE_1t/\hbar} + \phi_2(x) e^{-iE_2t/\hbar}\right)\!,$$
Where $$\phi_1(x)\,e^{-iE_1t/\hbar}$$ and $$\phi_2(x)\,e^{-iE_2t/\hbar}$$ are stationnary solutions of the Schrödinger equation with $$E_1 \neq E_2$$. By linearity, $$\psi$$ is also a solution of the Schrödinger equation and it cannot be factorized.
Now, since $$H$$ is Hermitian, it can be diagonalized and with some additional work you can show that the factorized solutions that you wrote form a complete basis of the solutions to Schrödinger equation, in the sense that any solution can be written as a linear combination of factorized solutions.