Energy Levels of Pionic and Kaonic Helium in the Variational Approach

In the variational method, we study the energy levels of pionic helium \((\pi -e-{\text{He}})\) and kaonic helium \((K - e - {\text{He}})\) with an electron in the ground state and a meson in the excited state with the principal and orbital quantum numbers n ~ l + 1 ~ 20. Variational wave functions are taken in the Gaussian form. Matrix elements of the basic Hamiltonian and corrections to vacuum polarization and relativism are calculated analytically in a closed form. We calculate some bound state energies and transition frequencies, which can be studied experimentally.