= Time-independent Schrödinger equation
{{wiki=Schrödinger_equation#Time-independent_equation}}
The <time-independent Schrödinger equation> is a variant of the <Schrödinger equation> defined as:
$$
\hat{H}[\psi_x(E, \vv{x})] = E \psi_x{\vv{x}}
$$
{title=Time-independent Schrodinger equation}
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: <Solving the Schrodinger equation with the time-independent Schrödinger equation>{full}.
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 <solutions of the Schrodinger equation>[when solving the Schrodinger equation], see e.g. <quantum harmonic oscillator>.