Three dimensional time-dependent Schrödinger equation is
Let us apply separation of variable
Plugging it in the time-dependent Schrödinger equation
Putting everything that depends on r, on right and putting everything that depends on t, on left
Since the left-hand side is dependent only on t and the right-hand side is dependent on r, the two side must be equal and constant. Let this constant be E ( Since right-hand side has dimensions of energy).
Putting E in equations
or
which is time independent Schrödinger equation.
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