equations:schroedinger_equation
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equations:schroedinger_equation [2020/04/10 11:40] 109.81.208.52 [Concrete] |
equations:schroedinger_equation [2020/11/21 01:43] (current) 2a01:cb15:33b:c600:f4a9:8015:b3fa:6b19 Typo in the Hamiltonian |
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| 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}$: | 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} \nabla^2 + \hat V \hat{=} \frac{\hat{p}^2}{2m} + \hat V. \end{equation} |
| 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. | ||
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| $$ \phi(t) = A e^{-Et/\hbar} $$ | $$ \phi(t) = A e^{-Et/\hbar} $$ | ||
| - | and the second equation is known as the stationary Schrödnger equation. This means that for all systems where the Hamiltonian does not explicitly depend on the time, we known immediately how the time-dependence of the total wave function $\Psi(x,t)$ looks like, namely: $\Psi(x,t) = \phi(t) \psi(x) = A e^{-Et/\hbar} \psi(x)$. The only thing we then have to do is to solve the __stationary Schrödinger equation__ | + | and the second equation is known as the stationary Schrödnger equation. This means that for all systems where the Hamiltonian does not explicitly depend on the time, we know immediately what the time-dependence of the total wave function $\Psi(x,t)$ looks like, namely: $\Psi(x,t) = \phi(t) \psi(x) = A e^{-Et/\hbar} \psi(x)$. The only thing we then have to do is to solve the __stationary Schrödinger equation__ |
| \begin{equation} | \begin{equation} | ||
equations/schroedinger_equation.1586511625.txt.gz · Last modified: 2020/04/10 09:40 (external edit)
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