A typical oscillating perturbation on protonium

Abstract

Abstract Protonium is a type of onium consisting of a proton (p) and an anti-proton ( p ¯ ) which is called antiprotonic hydrogen. Since the proton and the anti-particle have the same mass, and each of them is a fermion, the Hamiltonian form of this system consists of the kinetic energy terms of each particle, the potential energy, and the spin interactions of both (hyperfine structure). To simplify the analysis, the Hamiltonian is then converted into a Hamiltonian which contains the relative motion terms of the two particles. This new Hamiltonian contains their reduced mass because the two particles orbit each other towards a certain center of mass. In the initial state, t < 0, the system is in the ground state and then at 0 < t < τ, the system is given a disturbance of the form V ( t ) = ( V 0 + V 1 r sin θ e i ϕ ) cos ( ω t ) σ ^ p . x ^ , where V 0 is a constant, V 1 is a constant and very small, σ ^ p is the Pauli matrix for proton, and x ^ is the three-dimensional position operator in spherical coordinates. We analyze this system using a time-dependent perturbation theory that begins by defining the initial state, then calculating the transition amplitude for the new state when the perturbation is applied to the system. From the results of this calculation, we compute the system transition probability. We find that the transition probability of this system can only occur for the following quantum number terms, namely n even positive integer, l = 2,S = 1, and with the certain combinations of m and M.