From Wave Mechanics to Matrix Mechanics: The Finite Element Method

There are only a few analytic solutions available for this partial differential equation. Schrödinger found all of the most useful ones in his first paper on Quantum Mechanics: the Coulomb potential and the harmonic oscillator potential. He published the solution to the nonrelativistic Coulomb problem in his first Wave Mechanics paper but solved the relativistic problem six months earlier. He also solved the harmonic oscillator problem in his first paper and introduced what are now called coherent states in another paper six months afterward. With the advent of powerful computers and even more powerful linear algebra algorithms, partial differential equations are usually solved by first transforming them into matrix equations and then solving the matrix equations. Solutions almost always involve either matrix inversion or matrix diagonalization. The Schrödinger equation is usually not given in the original variational form. It is usually expressed as a second order partial differential equation. The relation between the two is obtained as follows. The kinetic energy term is integrated by parts: