Question

deduct the time-independent Schrödinger equation from the Schrödinger equation

Answer 1

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.