wave equation of a particle in a scalar potentialWaveEquationOfAParticleInAScalarPotential2009-04-05 16:05:492009-04-05 16:05:49Topic% this is the default PlanetMath preamble. as your knowledge
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% define commands hereIn order to form the wave equation of a particle in a potential $V(\mathbf{r})$, we operate at first under the conditions of the `geometrical optics approximation' and seek to form an equation of propagation for a wave packet $\Psi(\mathbf{r},t)$ moving in accordance with the de Broglie theory.
The center of the packet travels like a classical particle whose position, momentum, and energy we shall designate by $\mathbf{r}_{cl.}$, $\mathbf{p}_{cl.}$, and $E_{cl.}$, respectively. These quantities are connected by the relation
\begin{equation}
E_{cl.} = H(\mathbf{r}_{cl.},\mathbf{p}_{cl.}) = \frac{p^2_{cl.}}{2m} +V(\mathbf{r}_{cl.})
\end{equation}
$ H(\mathbf{r}_{cl.}$ is the classical Hamiltonian. We suppose that $V(\mathbf{r})$ does not depend upon the time explicitly (conservative system), although this condition is not absolutely necessary for the present argument to hold. Consequently $E_{cl.}$ remains constant in time, while $\mathbf{r}_{cl.}$ and $\mathbf{p}_{cl.}$ are well-defined functions of $t$. Under the approximate conditions considered here, $V(\mathbf{r})$ remains practically constant over a region of the order of the size of the wave packet; therefore
\begin{equation}
V(\mathbf{r}) \Psi(\mathbf{r},t) \approx V(\mathbf{r}_{cl.}) \Psi(\mathbf{r},t)
\end{equation}
On the other hand, if we restrict ourselves to time intervals sufficiently short so that the relative variation of $\mathbf{p}_{cl.}$ remains negligible, $\Psi(\mathbf{r},t)$ may be considered as a superposition of plane waves of the type
\begin{equation}
\Psi(\mathbf{r},t) = \int F(\mathbf{p}) \exp^{i(\mathbf{p} \cdot \mathbf{r} - Et)/\hbar} d\mathbf{p}
\end{equation}
whose frequencies are in the neighborhood of $E_{cl.}/\hbar$ and whose wave vectors lie close to $\mathbf{p}_{cl.}/\hbar$. Therefore
$$i\hbar \frac{\partial}{\partial t} \Psi(\mathbf{r},t) \approx E_{cl.} \Psi(\mathbf{r},t)$$
\begin{equation}
\frac{\hbar}{i} \nabla \Psi(\mathbf{r},t) \approx \mathbf{p}_{cl.}(t) \Psi(\mathbf{r},t)
\end{equation}
and taking the divergence of this last express ion, one obtains
\begin{equation}
- \hbar^2 \nabla^2 \Psi(\mathbf{r},t) \approx p^2_{cl.} \Psi(\mathbf{r},t)
\end{equation}
combining the relations (2),(3), and (4) and making use of equation (1), we obtain
$$
\i \hbar \frac{\partial}{\partial t} \Psi + \frac{\hbar^2}{2m} \nabla^2 \Psi - V \Psi \approx \left ( E_{cl.} - \frac{p^2_{cl.}}{2m} - V(\mathbf{r}_{cl.}) \right) \Psi \approx 0
$$
The wave packet $\Psi(\mathbf{r},t)$ satisfies - at least approximately - a wave equation of the type we are looking for. We are very naturally led to adopt this equation as the wave equation of a particle in a potential, and we postulate that in all generality, even when the conditions for the `geometrical optics' approximation are not fulfilled, the wave $\Psi$ satisfies the equation
\begin{equation}
i \hbar \frac{\partial }{\partial t} \Psi(\mathbf{r},t) = \left( -\frac{\hbar^2}{2m} \nabla^2 + V(\mathbf{r}) \right) \Psi(\mathbf{r},t)
\end{equation}
It is the Schr\"odinger equation for a particle in a potential $V(\mathbf{r})$.
[1] Messiah, Albert. "Quantum Mechanics: Volume I." Amsterdam, North-Holland Pub. Co.; New York, Interscience Publishers, 1961-62.
This entry is a derivative of the Public domain work [1].