Electric quadrupole potential energy for a particle with mass
is given by
Under this potential and transverse magnetic field
, the equations of motion for a charged particle are given by
The third equation is decoupled, thus the particle's motion is sinusoidal in the
direction. We can exactly solve the coupled
equations by introducing the complex variable
. In terms of
, we can write the first two equations as a single equation involving only
The general solution to this equation is
From this solution, we can see that the motion of the particle is stable (confined in a finite spatial region) when
For an antiproton, this condition corresponds roughly to
is measured in
 C. Gignoux and B. Silvestre-Brac, Solved Problems in Lagrangian and Hamiltonian Mechanics
, London: Springer, 2009.
 R. L. Tjoelker, "Antiprotons in a Penning Trap: A New Measurement of the Inertial Mass," Ph.D. thesis, Harvard University, Cambridge, 1990.