Stark Effects in Momentum Space

In this study, when hydrogen or a hydrogen-like atom is placed in a uniform electric field, shifts in energy levels are calculated in momentum space by using the Fourier transform of hydrogen type orbitals. By choosing the electric field in the z-direction, the z average value to be calculated to form the Hamiltonian matrix elements is written in the momentum space. The average value obtained is found as the product of the angular and radial integrals. To calculate the angular integrals, recurrence relations of the spherical harmonics especially in dipole transitions and derivative relations of spherical harmonics with respect to angles are used. The radial integrals are calculated analytically using the recurrence relations and series expressions of the Gegenbauer polynomials. The analytical expression obtained for the average value is quite convenient for numerical calculations. As a result, Hamiltonian matrix elements are obtained using the Mathematica programming language and the eigenvalues and eigenvectors of the matrix are calculated by using the Jacobi method. The results found are in complete agreement with the results in the configurations space.