equations:schroedinger_equation
Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision Next revision Both sides next revision | ||
|
equations:schroedinger_equation [2018/05/12 12:59] jakobadmin [Concrete] |
equations:schroedinger_equation [2019/05/21 16:55] michael remove duplicated line in free particle example |
||
|---|---|---|---|
| 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} | ||
| Line 42: | Line 42: | ||
| It is conventional to denote operators by an additional hat above the classical symbol. | It is conventional to denote operators by an additional hat above the classical symbol. | ||
| - | The Hamiltonian is what is different for different systems. Formulated differently, the Hamiltonian characterizes the system in question. The rest of the Schrödinger equation stays the same for all systems. | + | The Hamiltonian is what is different for different systems. Formulated differently, the Hamiltonian characterizes the system in question. The rest of the Schrödinger equation stays the same for all systems. For example, the Hamiltonian for a [[models:basic_models:harmonic_oscillator|harmonic oscillator]] reads |
| + | |||
| + | \begin{equation} \hat H \equiv \frac{\hat{p}^2}{2m} - \frac{1}{2}k \hat{x}^2 . \end{equation} | ||
| ---- | ---- | ||
| Line 82: | Line 85: | ||
| \begin{align} | \begin{align} | ||
| H \psi(x)&= E\psi(x) \notag \\ | H \psi(x)&= E\psi(x) \notag \\ | ||
| - | \frac{-\hbar \partial_x^2}{2m} \psi(x) &=E\psi(x) \notag \\ | ||
| \frac{-\hbar \partial_x^2}{2m} \psi(x) &=E\psi(x) \notag | \frac{-\hbar \partial_x^2}{2m} \psi(x) &=E\psi(x) \notag | ||
| \end{align} | \end{align} | ||
equations/schroedinger_equation.txt · Last modified: 2020/11/21 01:43 by 2a01:cb15:33b:c600:f4a9:8015:b3fa:6b19
Page Tools
Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 4.0 International
