Full analytical Expressions of the Quantum Mechanical Energy Eigenstates of charged Particles confined in a Penning Trap at non-relativistic Energies.

The storage of a few charged particles at low kinetic energy and at higher energies can be estimated by finding and characterizing the trajectories of charged particles which are classically bound within a Penning trap. However, there is benefit to using quantum mechanics to predict the energy levels and to map out the probability distributions of a few charged particles such as e`s and protons which are `trapped' in a given Penning Trap with particular settings for Bo (magnetic field) and quadrupole `E'strength (Vzz$_{\mathrm{[qua]}})$. We are able to analytically solve the Schr\"{o}dinger equation which represents the Hamiltonian of a single charged particle confined in a Penning `device'. Presuming that the L$_{\mathrm{[orbit]}}$ and `longitudinal' kinetic energies are non-relativistic, then the eigenvalues of the possible energies of the single charged particle follows a formula for ``QM'' integers $N$ and $m$: E$_{\mathrm{(N,m)}} \quad =$ 2*$\hbar $/$_{\mathrm{M}}$*G*(N$+$1) -m*$\hbar $*q*Bo/$_{\mathrm{M}}$, where G$=$ sqrt(q$^{\mathrm{2}}$*Bo$^{\mathrm{2}}$/4 -M/$_{\mathrm{2}}$*q*Vzz$_{\mathrm{[qua]}})$ . This work matches the formulation the famous QM text [1]. On the other hand, ref[1] allows B$_{\mathrm{o}}$ but not Vzz$_{\mathrm{[qua]}}$. \\ \\ Bibliography:\\ $[1]$ L. Landau and E. Lifshitz, "Quantum Mechanics – Course of Theoretical Physics", Volume 3-Third Edition, Pergamon Press, © 1977.