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equations:schroedinger_equation [2018/05/12 12:59] jakobadmin [Concrete] |
equations:schroedinger_equation [2019/05/21 16:53] michael minus sign and hbar missing from momentum operator |
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\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} | ||
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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 |
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+ | \begin{equation} \hat H \equiv \frac{\hat{p}^2}{2m} - \frac{1}{2}k \hat{x}^2 . \end{equation} | ||
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