This shows you the differences between two versions of the page.
Both sides previous revision Previous revision | Next revision Both sides next revision | ||
equations:schroedinger_equation [2018/05/14 07:05] jakobadmin [Concrete] |
equations:schroedinger_equation [2019/05/21 16:53] michael minus sign and hbar missing from momentum operator |
||
---|---|---|---|
Line 33: | Line 33: | ||
\begin{align} | \begin{align} | ||
- | \text{ the classical momentum } p_i \ &\rightarrow \ i \partial_{x_i} \, . | + | \text{ the classical momentum } p_i \ &\rightarrow \ {-i} \hbar \partial_{x_i} \, . |
\end{align} | \end{align} | ||
- | Formulated differently, the Hamiltonian operator is calculated from the classical energy $E= T +V$ by replacing the classical momentum $p_i$ with the momentum operator $ \hat{p}_i \equiv i \partial_{x_i}$: | + | Formulated differently, the Hamiltonian operator is calculated from the classical energy $E= T +V$ by replacing the classical momentum $p_i$ with the momentum operator $ \hat{p}_i \equiv {-i} \hbar \partial_{x_i}$: |
\begin{equation} \hat H \equiv - \frac{\hbar^2}{2m} \Delta^2 + \hat V \hat{=} \frac{\hat{p}^2}{2m} + \hat V. \end{equation} | \begin{equation} \hat H \equiv - \frac{\hbar^2}{2m} \Delta^2 + \hat V \hat{=} \frac{\hat{p}^2}{2m} + \hat V. \end{equation} |