Time-independent Schrödinger equation

The time-independent Schrödinger equation is a variant of the Schrödinger equation defined as:

So we see that for any Schrödinger equation, which is fully defined by the Hamiltonian $\hat{H}$, there is a corresponding time-independent Schrödinger equation, which is also uniquely defined by the same Hamiltonian.

The cool thing about the Time-independent Schrödinger equation is that we can always reduce solving the full Schrödinger equation to solving this slightly simpler time-independent version, as described at: Section "Solving the Schrodinger equation with the time-independent Schrödinger equation".

Because this method is fully general, and it simplifies the initial time-dependent problem to a time independent one, it is the approach that we will always take when solving the Schrodinger equation, see e.g. quantum harmonic oscillator.