應數筆記（勒壤得方程式，Legendre's Equation）

 

Schrodinger’s Function

Def.$$i\hbar\frac{\partial}{\partial t}|\Psi(t)\rangle=\hat{H}|\Psi(t)\rangle$$

If we consider a steady-state result, we have 

$$i\hbar\frac{\partial}{\partial t}|\Psi(t)|=\hat{H}|\Psi(t)\rangle\Rightarrow\frac{-\hbar^{2}}{2m}\nabla^{2}\Psi(\vec{r})+V(r)\Psi(\vec{r})=E~\Psi(\vec{r})$$  where V(r) is the potential and E is the energy of the object(s).