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Gauge invariance in single-particle non-relativistic QM

Here, we follow Merzbacher [2].

We now look at the interaction between charges and the electromagnetic field. The first step is to find the Hamiltonian, which is done via the sequence `classical Lagrangian - classical Hamiltonian - QM Hamiltonian'. For a particle of mass \bgroup\color{col1}$ m$\egroup with charge \bgroup\color{col1}$ q$\egroup in an electromagnetic field described by potentials \bgroup\color{col1}$ (\phi, \mathbf{A})$\egroup, the classical Hamiltonian is

$\displaystyle {\mathcal H} = H(t) + H_{\rm rad},\quad H(t)$ $\displaystyle \equiv$ $\displaystyle \frac{1}{2m}\left({\bf p}-q \mathbf{A}\right)^2 + q \phi$ (2.1)
$\displaystyle H_{\rm rad}$ $\displaystyle =$ $\displaystyle \frac{1}{2}\int d{\bf r} \left[ \varepsilon_0 \mathbf{E}_\perp^2 + \mu_0^{-1} \mathbf{B}^2\right].$ (2.2)

The em field are still treated classically here. The replacement of the momentum \bgroup\color{col1}$ {\bf p}$\egroup by
$\displaystyle {\bf p} \to {\bf p}-q \mathbf{A}$     (2.3)

is called minimal coupling. The term \bgroup\color{col1}$ H_{\rm rad}$\egroup is the energy of the electromagnetic field.



Subsections
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Next: Local Gauge Transformation Up: Interaction with Light Previous: Coulomb Gauge   Contents   Index
Tobias Brandes 2005-04-26