Wolfram Alpha:

```Schrodinger Wave Equation
-------------------------
Assume electron is a travelling wave (free particle) defined by:

ψ = Aexpi(kx-ωt))

Spatial derivative:

∂ψ/∂x = ikψ = (ip/h)ψ

Multiply both sides by ih to get:

-ih∂ψ/∂x = pψ

∴ (h2/2m)∂2ψ/∂x2 = (p2/2m)ψ

= Eψ   ... A.

This is the TIME INDEPENDENT form.

Time derivative:

In addition to its role in determining system energies, the Hamiltonian
operator also generates the time evolution of the wavefunction.

∂ψ/∂t = -iωψ = (-iE/h)ψ since E = hω

Multiply both sides by ih to get:

∴ ih∂ψ/∂t = Eψ  ... B.

Equating A. with B. we get:

-(h2/2m)∂2ψ/∂x2 = ih∂ψ/∂t

This is the TIME DEPENDENT form.

The time-independent Schrodinger equation is the equation describing
stationary states (atomic or molecular orbitals).  It is only used when
the Hamiltonian itself is not dependent on time.  Fortunately, the majority
of interesting problems in quantum mechanics involve stationary states
with definite total energy and so the time dependent form is not required.

Equation in the presence of a potential:

Time Independent form:

-(h2/2m)∂2ψ(x)/∂x2 + U(x)ψ(x) = Eψ(x)

Time Dependent form:

-(h2/2m)∂2ψ(x,t)/∂x2 + U(x)ψ(x,t) = ih∂ψ(x,t)/∂t```