As discussed in the article of time dependent Schrodinger wave equation:

V=A exp[-i/Ћ(Et-px]

= A exp(-i/Ћ Et) exp(i/Ћ)

Ψ=ψ^’ exp(-iEt/Ћ)                                                             (1)

Where Ћ = h/2π

So, ψ is a product of a time dependent function exp(-i/Ћ Et) and a position dependent function

Ψ^’= A exp(-i/Ћ px)

Differentiating equation (1) w.r.t.x, We have

dψ/dx = exp(-i/ЋEt) dψ^’/dx

and            d²ψ/dx²= exp(-i/Ћ Et) d²ψ^’/dx² (2)

Also on differentiating ψ w.r.t. t, we have

dψ/dt=ψ^’ exp (-iEt/Ћ) (I E/Ћ)

dψ/dt=-(iE/Ћ)ψ^’ exp(-I Et/Ћ)                                           (3)

Put equations [1-3] in time dependent Schrodinger wave equation (discussed earlier),

iЋ[-iE/Ћψ^’ exp(-iEt/Ћ)]= -Ћ²/2m[exp(i/ЋEt) d²ψ^’/dx²] +V ψ^’ exp(iEt/Ћ)

Eψ^’ exp(iEt/Ћ) = -Ћ²/2m exp(i/Ћ Et) d²ψ^’/dx² + V ψ^’exp(iEt/Ћ)

Dividing throughout by expression (i/Ћ Et) we have

Eψ^’= (-Ћ²/2π) d²ψ^’/dx²+V ψ^’

Or            (E-V)ψ^’=-Ћ²/2m  dψ^’/dx²

or             d²Ψ^’/dx² + (2m/Ћ²)(E-V)ψ^’ (4)

Which is time independent form of Schrodinger wave equation in one dimension.

In three-dimensional form:

d² Ψ^’/dx²+ d²ψ^’/dy²+ d²Ψ’/ d²x²+2m/Ћ²(E-V)ψ^’=0

In this equation, ψ’ equation, ψ’(x) is also called the wave function. The potential V(x) does not contain the time explicity and E, the total energy of the particle is a constant.