= Penning trap normal modes
{c}
{title2=$\omega_\pm=(\omega_c\pm\sqrt{\omega_c^2-2\omega_z^2})/2$}
Define positive <angular frequency> magnitudes $\omega_c=|Q|B/m$ and $\omega_z>0$, with uniform $\mathbf B=B\hat{\mathbf z}$, $B>0$. Choose the static electrode polarity through the <potential energy>
$$
Q\Phi=\frac{m\omega_z^2}{2}\left(z^2-\frac{x^2+y^2}{2}\right).
$$
Its axial <force> gives $\ddot z=-\omega_z^2z$, while its transverse <electric force> points outwards. For $Q>0$, let $w=x+iy$. The <Lorentz force> gives
$$
\ddot w+i\omega_c\dot w-\frac{\omega_z^2}{2}w=0.
$$
Trying $w\propto e^{-i\omega t}$ gives the two radial <normal mode> magnitudes $\omega_\pm$ displayed above: the fast modified <cyclotron> motion and slow <magnetron motion>. For $Q<0$ the rotation directions reverse. Bounded confinement in this ideal model requires $\omega_c^2>2\omega_z^2>0$; the repeated-root boundary needs exclusion because it allows a linearly growing solution. At large $\omega_c/\omega_z$, $\omega_-\simeq\omega_z^2/(2\omega_c)$ and $\omega_+\simeq\omega_c-\omega_-$.
The ideal identities
$$
\omega_++\omega_-=\omega_c,\qquad
\omega_+\omega_-=\frac{\omega_z^2}{2},\qquad
\omega_+^2+\omega_-^2+\omega_z^2=\omega_c^2
$$
follow directly from that quadratic. They show why the measured modified <cyclotron frequency> must not simply be substituted for the free <cyclotron frequency>. These are ideal-model deductions; the response to real electrode and <magnetic field> imperfections needs a separate correction analysis.
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