Incident ¹H⁺ and ⁴He⁺ ions at 0.3-2.6 MeV and ¹⁹F^q⁺ ions at 25, 27 and 35 MeV were used to study the M-shell x-ray production cross sections of Au, Pb, Bi and U. For the incident fluorine ions, projectile charge state dependence of the cross sections were extracted from measurements made with varying target thicknesses ( ~1 to ~300 μg/cm²). The efficiency of the Si(Li) detector was determined by measuring the K-shell x-ray production of various low Z elements and comparing these values to the prediction of the CPSS theory. The experimental results are compared to the prediction of first Born approximation for direct ionization to the continuum and to the OBK of Nikolaev for the electron capture to the K-, L-, M-...shells of the incident ion. Comparison is also made with the ECPSSR theory that accounts for the energy loss, Coulomb deflection, and relativistic effects in the perturbed stationary state theory.

A system of nonrelativistic charged particles and radiation is canonically quantized in the Coulomb gauge and Maxwell's equations in quantum electrodynamics are derived. By requiring form invariance of the Schrodinger equation under a space and time dependent unitary transformation, operator gauge transformations on the quantized electromagnetic potentials and state vectors are introduced. These gauge transformed potentials have the same form as gauge transformations in non-Abelian gauge field theories. A gauge-invariant method for solving the time-dependent Schrodinger equation in quantum electrodynamics is given. Maxwell's equations are written in a form which holds in all gauges and which has formal similarity to the equations of motion of non-Abelian gauge fields. A gauge-invariant derivation of conservation of energy in quantum electrodynamics is given. An operator gauge transformation is made to the multipolar gauge in which the potentials are expressed in terms of the electromagnetic fields. The multipolar Hamiltonian is shown to be the minimally coupled Hamiltonian with the electromagnetic potentials in the multipolar gauge. The model of a charged harmonic oscillator in a single-mode electromagnetic field is considered as an example. The gauge-invariant procedure for solving the time-dependent Schrodinger equation is used to obtain the gauge-invariant probabilities that the oscillator is in an …