The Schroedinger Equation serves as one of the most improtant and utilized equations in all of quantum mechanics. The basics of the formula all stem from the tradtional laws of conservation of energy, and allows for an exceptional amount of predictability in the movemements of wavefunctions. The most common form of the equation as seen in quantum mechanics is listed below.–
This particular derivation is only pertaining to the two contributors to the total energy of a system, kinetic and potential. The toal of these two will also sum and take the form of the Hamiltonian Operator ( Hψ ) when applied to the Simple Harmonic Oscillator (S.H.O).
Some other applications of this equation are the particle in a box and calculations of energy states. Below is linked a helpful video to investigate this further.
Credit: University of Victoria DB., Jan Jensen: University of Copenhagen