Schrödinger's Equation (idea)

-((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:

• h - Planck's Constant - Usually (h/2π) is called hbar, and has a value of 0.6582*10^-15 eV·s
• m -mass of the particle being examined.
• Ψ - The wave function. This is what is usually being computed using Schroedinger's Equation.
• V - This is the potential energy of the described particle.
• j - The imaginary number, being equal to √-1.
• x - Position
• t - Time

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