Basic wave mechanics

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 $$ \vec p $$. The relation $$ \frac{{|\vec p{|^2}}}{{2\mu }}\;{\rm{ + V = E}} $$ is transcribed into quantum mechanics by substituting operators for the physical observables total energy, momentum, and potential according to: $$ E\; \to \;E\_{\rm{ = ih - }}\;\frac{\partial }{{\partial t}},\;{\rm{\vec p}}\; \to \;{\rm{\vec p\_ = - ih - }}\vec \nabla \_,\;{\rm{V }} \to {\rm{ V\_}}{\rm{.}} $$