with an additional confining potential of the form $m\omega_0^2r^2/2$, the Hamiltonian given in Eq. (\[ham\_symm1\]) becomes $$H=\frac{{\mathbf{p}}^2}{2m}+\frac{\omega_c}{2}L_z+\frac{1}{2}m\Omega^2r^2,
\label{ham_parabolic}$$ where $\Omega \equiv \sqrt{\omega_0^2+\omega_c^2/4}$. With this parabolic confinement, the eigenenergies are $$E_{n_r,l}=2\hbar\Omega \biggl(n_r + \frac{1}{2} [ |\ell | + \frac{\omega_c}{2 \Omega} \ell] + \frac{1}{2} \biggr).
    \label{energies_parabolic}$$ Figure \[fock\_darwin\] shows a typical Fock-Darwin plot. A variation on this plot is shown in Fig. (\[fock\_darwin\_fm\]). As in Fig. \[fock\_darwin\], Eq. (\[energies\_parabolic\]) is used for the energies, but now they are normalized to the energy $\hbar \omega_c$, and are plotted as a function of the (normalized) confinement potential.

![Fock-Darwin spectra obtained from using the symmetric gauge in a parabolic trap, i.e. Eq. (\[energies\_parabolic\]). The crossing of energy levels is the defining feature of the Fock-Darwin spectra. The dashed lines correspond to the free Landau levels. Note however, that there is no condensation of levels as the field increases, in contrast to what happens with more confining traps, to be discussed later. []{data-label="fock_darwin"}](FD-spec-fig-2.pdf)

![The same Fock-Darwin spectra obtained as in the previous figure by confining the electron with a parabolic trap, in the Symmetric gauge, but now $E_{n_r,\ell}/(\hbar \omega_c)$ is plotted vs $\omega_0/\omega_c$ for some selected levels. The numbers in brackets label the $(n_r,\ell)$ quantum numbers. 