Why do we need both, the time-dependent and time-independent Schrödinger Equations?
Answer
Your question implies that the time-dependent and time-independent equations are somehow different things, but while technically true this is misleading. The time-independent form is a simplification we can make when the potential does not change with time, so if we start with the time-dependent equation we get the time-independent form for free.
As for why we use the time-independent equation: speaking as an (ex) chemist we're frequently interested in calculating the energy of molecules that are not changing with time. The time-independence means we can use the time-independent Schrödinger equation and it's quicker and easier to use i.e. less CPU time and memory needed to do the calculation.
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