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

Here we repeat the arguments from the [[../WaveEquationOfAParticleInAScalarPotential/|wave equation of a particle in a scalar potential]] and extend it to a more general case where the potential $V$ is an explicit [[../Bijective/|function]] of time, specifically a [[../Particle/|particle]] with [[../Charge/|charge]] $e$ in an electromagnetic [[../CosmologicalConstant/|field]] derived from a [[../SolenoidalVectorField/|vector potential]] $𝐀(𝐫,t)$ and a [[../Vectors/|scalar]] potential $\varphi (𝐫,t)$. In the latter case, the classical [[../Bijective/|relation]]

$E_{cl.}=H({𝐫}_{cl.},{𝐩}_{cl.})=\frac{p_{cl.}^2}{2m}+V({𝐫}_{cl.})$

must be replaced by the relation

$E=\frac{1}{2m}{(𝐩-\frac{e}{𝐀}(𝐫,t))}^2+e\varphi (𝐫,t).$

Considerations of the behavior of [[../CosmologicalConstant2/|wave]] packets on the "geometrical optics" approximation lead us to the [[../TransversalWave/|wave equation]]

$i\hslash \frac{\partial }{\partial t}\mathrm{\Psi }(𝐫,t)=[\frac{1}{2m}{(\frac{\hslash }{i}\mathrm{\nabla }-\frac{e}{𝐀})}^2+e\varphi ]\mathrm{\Psi }(𝐫,t)$

It is the Schr\"odinger equation of a charged particle in an electromagnetic field. On the right hand side of equation (3), the [[../QuantumOperatorAlgebra4/|operator]] ${(\frac{\hslash }{i}\mathrm{\nabla }-\frac{e}{𝐀})}^2$

designates the [[../DotProduct/|scalar product]] of the [[../Vectors/|vector]] operator $\frac{\hslash }{i}\mathrm{\nabla }-\frac{e}{𝐀}$ by itself; in other words, the function which results from its action on $\mathrm{\Psi }$ is the sum of the expression

$(\frac{\hslash }{i}\frac{\partial }{\partial x}-{\frac{e}{A}}_x)(\frac{\hslash }{i}\frac{\partial }{\partial x}-{\frac{e}{A}}_x)\mathrm{\Psi }=-{\hslash }^2\frac{{\partial }^2\mathrm{\Psi }}{\partial x^2}-\frac{e\hslash }{ic}(A_x\frac{\partial \mathrm{\Psi }}{\partial x}+\frac{\partial }{\partial x}(A_x\mathrm{\Psi }))+\frac{e^2}{c^2}{A}_x^2\mathrm{\Psi }$

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

$-{\hslash }^2{\mathrm{\nabla }}^2\mathrm{\Psi }-2\frac{e\hslash }{ic}(𝐀\cdot \mathrm{\nabla }\mathrm{\Psi })+(-\frac{e\hslash }{ic}(\mathrm{\nabla }\cdot 𝐀)+\frac{e^2}{c^2}{A}^2)\mathrm{\Psi }$

In all of this one must realize that the components of the operator $\mathrm{\nabla }$ and those of the operator $𝐀$ do not in general [[../Commutator/|commute]] with each other.

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

$i\hslash \frac{\partial }{\partial t}\mathrm{\Psi }(𝐫,t)=(-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2+V(𝐫))\mathrm{\Psi }(𝐫,t)$

and equation (3) are the generalizations of the [[../WaveEquationOfAFreeParticle/|wave equation of a free particle]] and the same remarks apply to them. They are indeed linear, homogeneous, [[../DifferentialEquations/|partial differential equations]] of the first order in the time. Furthermore, they can be deduced from the classical relations by the correspondence relation

$E\to i\hslash \frac{\partial }{\partial t},𝐩\to \frac{\hslash }{i}\mathrm{\nabla }$

References

[1] Messiah, Albert. "[[../QuantumParadox/|Quantum mechanics]]: [[../Volume/|volume]] I." Amsterdam, North-Holland Pub. Co.; New York, Interscience Publishers, 1961-62.

This entry is a derivative of the Public [[../Bijective/|domain]] [[../Work/|work]] [1].