Talk:PlanetPhysics/Wave Equation of a Particle in a Scalar Potential

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: wave equation of a particle in a scalar potential %%% Primary Category Code: 03.65.-w %%% Filename: WaveEquationOfAParticleInAScalarPotential.tex %%% Version: 2 %%% Owner: bloftin %%% Author(s): bloftin %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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In order to form the \htmladdnormallink{wave equation}{http://planetphysics.us/encyclopedia/TransversalWave.html} of a \htmladdnormallink{particle}{http://planetphysics.us/encyclopedia/Particle.html} 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 \htmladdnormallink{wave}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} packet $\Psi(\mathbf{r},t)$ moving in accordance with the de Broglie theory.

The center of the packet travels like a classical particle whose \htmladdnormallink{position}{http://planetphysics.us/encyclopedia/Position.html}, \htmladdnormallink{momentum}{http://planetphysics.us/encyclopedia/Momentum.html}, and \htmladdnormallink{energy}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} we shall designate by $\mathbf{r}_{cl.}$, $\mathbf{p}_{cl.}$, and $E_{cl.}$, respectively. These quantities are connected by the \htmladdnormallink{relation}{http://planetphysics.us/encyclopedia/Bijective.html} \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.}, \mathbf{p}_{cl.})$ is the classical \htmladdnormallink{Hamiltonian}{http://planetphysics.us/encyclopedia/Hamiltonian2.html}. We suppose that $V(\mathbf{r})$ does not depend upon the time explicitly (conservative \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/GenericityInOpenSystems.html}), 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 \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} 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 \htmladdnormallink{type}{http://planetphysics.us/encyclopedia/Bijective.html} \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 \htmladdnormallink{vectors}{http://planetphysics.us/encyclopedia/Vectors.html} 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 \htmladdnormallink{divergence}{http://planetphysics.us/encyclopedia/DivergenceOfAVectorField.html} 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. "\htmladdnormallink{Quantum mechanics}{http://planetphysics.us/encyclopedia/QuantumParadox.html}: \htmladdnormallink{volume}{http://planetphysics.us/encyclopedia/Volume.html} I." Amsterdam, North-Holland Pub. Co.; New York, Interscience Publishers, 1961-62.

This entry is a derivative of the Public \htmladdnormallink{domain}{http://planetphysics.us/encyclopedia/Bijective.html} \htmladdnormallink{work}{http://planetphysics.us/encyclopedia/Work.html} [1].

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