Minimal Coupling—Non-Relativistically

Exercise 2.5 Show that the electric field r,i) and the magnetic induction in Eqs. 

Hint Recall that the curl of a gradient vanishes, i.e. V x Vx(r, t) = 0. 

Using the scalar and vector potential we can write the expression for the Lorentz force alternatively as 

The usual way to treat the interaction between electromagnetic fields or nuclear electromagnetic moments and molecules is a semi-classical way, where the fields or nuclear moments are treated classically and the electrons are treated by quantum mechanics. The fields or nuclear moments are thus not part of the system, which is treated quantum mechanically, but they are merely considered to be perturbations that do not respond to the presence of the molecule. They therefore enter the molecular Hamiltonian in terms of external potentials similar to the Coulomb potential due to the charges of the nuclei. This is therefore called the minimal coupling approach. 

In order to reduce the number of indices and summation sign we derive here the Hamiltonian operator for the motion of a single electron in the presence of external fields. The final equations can then easily be generalized to the many-electron case in Section 2.8. 

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