Schroedinger's Equation is:
-((h/2π)2/2m)(∂2Ψ(x,t)/∂x2)+V(x,t)Ψ(x,t) = j(h/2π)(∂Ψ(x,t)/∂t)
A description of each variable:
The essence of Schroedinger's Equation is that it sums the energy for the particle, ala the classical physics equation E = K + V, where E is the total energy, K is the kinetic energy, and V is the potential energy.
In Schroedinger's Equation E is expressed in the right-hand side of the equation, V is expressed in V(x,t)Ψ(x,t), and K is expressed in -((h/2π)2/2m)(∂2Ψ(x,t)/∂x2)
This is the time-dependent Schroedinger Equation. There also exists a time-independent Schroedinger Equation, which easily derived from the time-dependent version via the technique of separation of variables (using Ψ(x,t)=ψ(x)φ(t)), and which can be used if the potential energy of the particle does not vary with time. It is:
-((h/2π)2/2m)(∂2ψ(x)/∂x2)+V(x)ψ(x) = Eψ(x)
Which can be rewritten, by factoring out ψ from the left-hand side, and using the Hamiltonian operator (H):
Hψ = Eψ