Talk:PlanetPhysics/Wave Equation of a Charged a Particle in an Electromagnetic Field

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: wave equation of a charged a particle in an electromagnetic field %%% Primary Category Code: 03.65.-w %%% Filename: WaveEquationOfAChargedAParticleInAnElectromagneticField.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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Here we repeat the arguments from the \htmladdnormallink{wave equation of a particle in a scalar potential}{http://planetphysics.us/encyclopedia/WaveEquationOfAParticleInAScalarPotential.html} and extend it to a more general case where the potential $V$ is an explicit \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} of time, specifically a \htmladdnormallink{particle}{http://planetphysics.us/encyclopedia/Particle.html} with \htmladdnormallink{charge}{http://planetphysics.us/encyclopedia/Charge.html} $e$ in an electromagnetic \htmladdnormallink{field}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} derived from a \htmladdnormallink{vector potential}{http://planetphysics.us/encyclopedia/SolenoidalVectorField.html} $\mathbf{A}(\mathbf{r},t)$ and a \htmladdnormallink{scalar}{http://planetphysics.us/encyclopedia/Vectors.html} potential $\phi(\mathbf{r},t)$. In the latter case, the classical \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}

must be replaced by the relation

\begin{equation} E = \frac{1}{2m} \left( \mathbf{p} - \frac{e}{c} \mathbf{A}(\mathbf{r},t) \right)^2 + e \phi(\mathbf{r},t). \end{equation}

Considerations of the behavior of \htmladdnormallink{wave}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} packets on the "geometrical optics" approximation lead us to the \htmladdnormallink{wave equation}{http://planetphysics.us/encyclopedia/TransversalWave.html} \begin{equation} i\hbar \frac{\partial}{\partial t} \Psi(\mathbf{r},t) = \left [ \frac{1}{2m} \left ( \frac{\hbar}{i} \nabla - \frac{e}{c} \mathbf{A} \right)^2 + e\phi \right] \Psi(\mathbf{r},t) \end{equation}

It is the Schr\"odinger equation of a charged particle in an electromagnetic field. On the right hand side of equation (3), the \htmladdnormallink{operator}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html} $$\left( \frac{\hbar}{i} \nabla - \frac{e}{c} \mathbf{A} \right )^2$$

designates the \htmladdnormallink{scalar product}{http://planetphysics.us/encyclopedia/DotProduct.html} of the \htmladdnormallink{vector}{http://planetphysics.us/encyclopedia/Vectors.html} operator $\frac{\hbar}{i} \nabla - \frac{e}{c} \mathbf{A}$ by itself; in other words, the function which results from its action on $\Psi$ is the sum of the expression

$$ \left( \frac{\hbar}{i} \frac{\partial}{\partial x} - \frac{e}{c} A_x \right ) \left ( \frac{\hbar}{i} \frac{\partial}{\partial x} - \frac{e}{c} A_x \right) \Psi = -\hbar^2 \frac{\partial^2 \Psi}{\partial x^2} - \frac{e\hbar}{ic} \left( A_x \frac{\partial \Psi}{\partial x}+ \frac{\partial}{\partial x} (A_x \Psi) \right ) + \frac{e^2}{c^2} A_x^2 \Psi$$

and of two other expressions which are obtained from it by substituting $y$ and $z$ for $x$, namely

$$ -\hbar^2 \nabla^2 \Psi - 2 \frac{e\hbar}{ic}(\mathbf{A} \cdot \nabla \Psi) + \left ( -\frac{e\hbar}{ic} (\nabla \cdot \mathbf{A}) + \frac{e^2}{c^2}A^2 \right) \Psi $$

In all of this one must realize that the components of the operator $\nabla$ and those of the operator $\mathbf{A}$ do not in general \htmladdnormallink{commute}{http://planetphysics.us/encyclopedia/Commutator.html} with each other.

The Schr\"odinger equation for a particle in a potential $V(\mathbf{r})$,

\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}

and equation (3) are the generalizations of the \htmladdnormallink{wave equation of a free particle}{http://planetphysics.us/encyclopedia/WaveEquationOfAFreeParticle.html} and the same remarks apply to them. They are indeed linear, homogeneous, \htmladdnormallink{partial differential equations}{http://planetphysics.us/encyclopedia/DifferentialEquations.html} of the first order in the time. Furthermore, they can be deduced from the classical relations by the correspondence relation

$$ E \rightarrow i \hbar \frac{\partial}{\partial t}, \,\,\,\,\,\, \mathbf{p} \rightarrow \frac{\hbar}{i}\nabla$$

\subsection{References}

[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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