PX262 - F1 - the schrödinger equation in 3D

time-dependent

• the time-dependent schrödinger equation in 3D is:$$\begin{array}{}\text{(1)}& i\hslash \frac{\partial \psi }{\partial t}=-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2\psi +V(\overrightarrow{r})\psi =\hat{H}\psi \end{array}$$where,
  ${\mathrm{\nabla }}^2=(\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}+\frac{{\partial }^2}{\partial z^2})$ ,
  $\hat{H}=(-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2+\hat{V})$ is the hamiltonian operator

time-independent

• considering the cases of time-independent potential, $V(\overrightarrow{r})$
• using separation of variables in the wavefunction:$$\psi (\overrightarrow{r},t)=\mu (\overrightarrow{r})T(t)$$
• substituting it into equation $(1):$

• the time-dependent part is:$$\frac{i\hslash }{T(t)}\frac{\partial T(t)}{\partial t}=E⟹T(t)=\exp (-\frac{iEt}{\hslash })$$

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