Basic wave mechanics

Abstract

The Schrödinger equation is based on the classical relationship between the total energy E, the kinetic energy $T = \frac{{{p^2}}}{{2\mu }}$ and the potential energy V, T + V = E, of a particle of mass μ and momentum $\overrightarrow p$. The relation $\frac{{{{\left| {\overrightarrow p } \right|}^2}}}{{2\mu }} + V = E$ is transcribed into quantum mechanics by substituting operators for the physical observables total energy, momentum, and potential energy according to:

$$E \to \underline E = i\hbar \frac{\partial }{{\partial t}}$$

,

$$\overrightarrow p \to \underline {\overrightarrow p } = - i\hbar \underline {\overrightarrow \nabla }$$

,

$$V \to \underline V$$