Exact Transition Matrix Elements Research Articles

EVALUATION OF RELATIVISTIC TRANSITION MATRIX ELEMENTS OF HYDROGENIC ATOMS USING PLANE WAVE EXPANSION

For the first time to our knowledge, a general, explicit formula for exact transition matrix elements in relativistic hydrogenic atoms is derived, by using the plane-wave expansion for the electromagnetic-field vector potential. By applying the obtained formula, discrete-discrete and discrete-continuous matrix elements are evaluated analytically and numerically.

Simplified diagrammatic expansion for effective operators.

For a quantum many-body problem, effective Hamiltonians that give exact eigenvalues in reduced model space usually have different expressions, diagrams, and evaluation rules from effective transition operators that give exact transition matrix elements between effective eigenvectors in reduced model space. By modifying these diagrams slightly and considering the linked diagrams for all the terms of the same order, we find that the evaluation rules can be made the same for both effective Hamiltonian and effective transition operator diagrams, and in many cases it is possible to combine many diagrams into one modified diagram. We give the rules to evaluate these modified diagrams and show their validity.

Evaluation of relativistic transition matrix elements and their contributions to the discrete-spectrum Lamb shift in hydrogenic atoms

For the first time to our knowledge, a general formula for exact transition matrix elements relativistic hydrogenic atoms is derived. By using the obtained formula the matrix elements for different electric and magnetic multipole transitins are evaluated analytically and numerically. Finally, by applying the exact transition matrix elements the contributions of individual discrete states and those of the whole discrete spectrum to the Lamb shift of the 1 s state are explicitly calculated.

RELATIVISTIC TRANSITION MATRIX ELEMENTS IN THE CASE OF HYDROGENIC ATOMS

A general formula for exact transition matrix elements in relativistic hydrogenic atoms is derived. By using the obtained formula an explicit evaluation for different electric dipole transition matrix elements is carried out analytically and numerically.

Analytic evaluation of exact transition matrix elements in nonrelativistic hydrogenic atoms

By carrying out additional integrations in the general formula for exact transition matrix elements in nonrelativistic hydrogenic atoms, which has been derived by Moses (Phys. Rev. A 8 1710 (1973)), significant simplifications are achieved. By using the obtained reduced formula, for the first time to our knowledge, the exact matrix elements for different electric and magnetic multipole transitions are evaluated analytically.

Calculation of the discrete-spectrum Lamb-shift contribution in atomic hydrogen

By applying the exact transition matrix elements, which were evaluated in our preceding paper, unrenormalized and renormalized Lamb-shift contributions stemming from individual discrete states as well as from the whole discrete spectrum are calculated in the case of the states 1S, 2S and 2P. For the first time to our knowledge, it is pointed out that the unrenormalized discrete spectrum contribution is completely finite and thus conceptual clarifications in the renormalization procedure are achieved.

Non-relativistic exact transition matrix elements for hydrogenic atoms

By using a general formula due to Moses (Phys. Rev. A,8, 1710 (1973)) exact matrix elements for different transitions in non-relativistic amomic hydrogen are evaluated explicitly.

UNRENORMALIZED AND RENORMALIZED DISCRETE-SPECTRUM LAMB-SHIFT CONTRIBUTIONS FOR 1S AND 2S HYROGENIC STATES

A general formula due to Moses (Phys. Rev. A8, 1710 (1973)) for exact transition matrix elements in nonrelativistic hydrogenic atoms has been reduced by carrying out additional calculations. By using the obtained result, unrenormalized (finite) and renormalized Lamb-shift contributions stemming from individual discrete states as well as from the whole discrete spectrum are calculated.

Theory of Dissociative Attachment

A fairly comprehensive theoretical treatment of the dissociative attachment (DA) of electrons to diatomic molecules ($AB+e\ensuremath{\rightarrow}{A}^{\ensuremath{-}}+B$) is given, going from general formalism to explicit cross-section formulas and comparison with experimental results. Using the Born-Oppenheimer separation, the process is understood as an electronic transition from a continuum to a discrete electronic state, which then dissociates. The final discrete state, being degenerate with a continuum, is necessarily a resonance. Accordingly, the theory is derived from a general rearrangement formalism which is based on projection operators onto this resonance state (defined along the lines of Feshbach) and uses the Born-Oppenheimer separation. This formalism, which is applicable to a fairly wide class of collision processes, is characterized by its extreme simplicity and practical usefulness. Exact transition matrix elements are derived for DA, together with its inverse and competing processes, in terms of the resonance and wave functions. Finally, a certain adiabatic approximation is considered for the potential scattering function, which neglects the indirect influence of other inelastic processes on the resonance transition, treating them as higher order effects. After this approximation, the classical nature of the nuclear motion makes it possible to eliminate the nuclear wave functions in a fairly trivial way and to derive a simple explicit formula for the DA cross section from an arbitrary initial vibration-rotational state of the molecule. For the most interesting case, the ground vibrational and rotational state, this formula is in complete agreement with the previous result of Bardsley, Herzenberg, and Mandl. Later this adiabatic approximation is partly relaxed and more general formulas derived. Various implications of the results are also considered, such as the relation to experimentally observed isotope effects and temperature effects. The vertical onset threshold phenomenon which occurs when the resonance potential curve is attractive is also discussed.