(6212) Hydrogen atom in magnetic field, probed by electric field
Consider an Hydrogen atom that is placed in a constant homogeneous magnetic field
\( \mathcal{B} \)
that points in the Z direction.
An electric field
\( \mathcal{E} \)
is turned on.
The interaction term is
\( V = - e\mathcal{E} \cdot r \)
.
The spin of the electron can be ignored.
(1) Calculate the matrix elements of
\( V \)
in the standard basis for
\( \mathcal{E} \)
that points in the Z direction.
(2) Calculate the matrix elements of
\( V \)
in the standard basis for
\( \mathcal{E} \)
that points in the X direction.
(3) Repeat the derivation in the latter case using coordinates such that
\( \mathcal{B} \)
points in the X direction, and
\( \mathcal{E} \)
in the Z direction.
(4)
Explain what are the selection rules of transitions in the following cases:
\( \mathcal{E} = (0,0,\cos(\Omega t)) \)
\( \mathcal{E} = (\cos(\Omega t),0,0) \)
\( \mathcal{E} = (\cos(\Omega t),\sin(\Omega t),0) \)
Assume that
\( \Omega \)
is comparable to the Zeeman splitting.