What does the Schrödinger equation tell you?

The Schrödinger equation is one of many similar equations in quantum mechanics. Each of these equations tells you for a particular physical property (a coordinate, angular momentum, ...) what wave functions have a well-defined value for that property. The Schrödinger equation gives the wave functions with well-defined values for the energy of a system. It would be wrong, however, to think that the system cannot be in another wave function. If it is in such other wave function, it then only doesn't have a definite energy. More precisely, if you would measure the energy of a system a number of times with the system each time in the same wave function that is not a solution of the Schrödinger equation, then it would be possible to get different results for the different measurements. If the system is in a wave function that is a solution of the Schrödinger equation, then you would always get the corresponding energy from the Schrödinger equation.

There are two reasons why the Schrödinger equation is special. The first reason has to do with the importance of the energy. A system (certainly an atom or molecule) is generally not isolated, but in thermal contact with its environment. It then has a temperature, and the probability to find the system with a certain energy is proportional to the Boltzmann factor exp[-E/RT]. For the electronic wave function of an atom or molecule the difference in energy for the solutions of the Schrödinger equation are normally so large that the atom or molecule is always in the ground state; i.e., the solution of the Schrödinger equation with the lowest energy. The energy is also important because in many spectroscopic techniques we see energy differences.

The second reason for the importance of the Schrödinger equation is that its solutions do not vary in time. They are therefore also called stationary states. Once a system is in a stationary state it will stay in that state until an external perturbation changes the wave function. Wave functions that are not solutions of the Schrödinger equation change in time.