A time-dependent perturbation theory which is based upon the U-matrix approach is presented using the interaction e-αt(1 + e-βt)−1V(x) with β real, α complex and β > Re α > 0. Thus the adiabatic or time-independent approximation (β → 0) and the sudden approximation (β → ∞) can be obtained using just one formalism. The usual series for the U matrix is derived and shown to converge for a semi infinite range if the interaction is of the form |t|-δ, δ > 1, for large times t. Two interesting results are (1) The derivation of the ``golden rule'' from the discrete-state time-dependent perturbation theory presented in most text books of quantum mechanics leads to erroneous physical interpretations because the energy-level shift is ignored. (2) When considering scattering between states in the continuum, it is found that a characteristic feature of time-dependent interactions is a discrete momentum spectrum of final states which, in the relativistic case, leads to a mass spectrum. These spectra cannot be obtained using the S matrix.
Within density-functional theory a time-dependent perturbation theory is derived that yields exact orbital-dependent expressions for the time-dependent exchange-correlation potential in the form of a perturbation series. This leads to an exact time-dependent Kohn-Sham scheme based on perturbation theory. In first order an ``exchange-only'' Kohn-Sham procedure is obtained, that is a generalization of a recently presented time-dependent optimized potential method. The exact, in higher order, but quite complicated expressions for the exchange-correlation potential may serve as a starting point for the construction of approximate exchange-correlation potentials. An adiabatic connection or coupling constant path between a time-dependent physical system and its corresponding Kohn-Sham system is introduced.
We introduce a method of evaluating the density matrix of an arbitrary quantum mechanical system in terms of the quantities pertinent to the solution of the time-dependent density functional theory (TDDFT) problem. Our theory utilizes the adiabatic connection perturbation method [G\{o}rling and Levy, Phys. Rev. A {\bf 50}, 196 (1994), G\{o}rling, ibid. {\bf 55}, 2630 (1997)], from which the expansion of the density matrix in powers of the interaction constant
λ naturally arises. By this, we obtain the one-density-matrix $\rho_\lambda(\rv,\rv',t)$, which, by construction, has the
λ-independent diagonal elements $\rho_\lambda(\rv,\rv,t)=n(\rv,t)$, where $n(\rv,t)$ is the particle density. The off-diagonal elements of $\rho_\lambda(\rv,\rv',t)$ importantly contribute to the processes beyond the reach of TDDFT, of which we consider the momentum-resolved photoemission, doing this to the first order in
λ (exact exchange). In an illustrative calculation of photoemission from the quasi-2D electron gas with {\it one filled subband} we find strong deviations from the independent-particle Fermi golden rule formula.
An alternative to the Magnus expansion in time-dependent quantum mechanical perturbation theory is presented. An exponential form of the time-development operator, given as the exponential of a sum of products of integrals well known in time-dependent quantum mechanics, is derived from the standard perturbation expansion. The derivation is simple and the form of the terms in the exponential is so straightforward that the Nth order term can be written by inspection. The standard Magnus terms can be obtained by rearrangement of the terms of the new expansion probably easier than they can be obtained by iteration of the original equations. A proof by induction is presented to establish the form of the Nth term and it is shown how the new expansion terms relate to the Magnus terms. Some apparent differences between Magnus terms in the literature are resolved.
We evaluate the density matrix of an arbitrary quantum mechanical system in terms of the quantities pertinent to the solution of the time-dependent density functional theory (TDDFT) problem. Our theory utilizes the adiabatic connection perturbation method of G\"orling and Levy, from which the expansion of the many-body density matrix in powers of the coupling constant $\ensuremath{\lambda}$ naturally arises. We then find the reduced density matrix ${\ensuremath{\rho}}_{\ensuremath{\lambda}}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}},t)$, which, by construction, has the $\ensuremath{\lambda}$ independent diagonal elements ${\ensuremath{\rho}}_{\ensuremath{\lambda}}(\mathbf{r},\mathbf{r},t)=n(\mathbf{r},t)$,
n(r,t) being the particle density. The off-diagonal elements of ${\ensuremath{\rho}}_{\ensuremath{\lambda}}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}},t)$ contribute importantly to the processes unaccessible via the density, directly or by the use of the known TDDFT functionals. Of those, we consider the momentum-resolved photoemission, doing this to the first order in $\ensuremath{\lambda}$, i.e., on the level of the exact exchange theory. In illustrative calculations of photoemission from the quasi-2D electron gas and isolated atoms, we find quantitatively strong and conceptually far-reaching differences with the independent-particle Fermi's golden rule formula.
We now consider systems for which the Hamiltonian contains time-dependent interaction terms. In such cases we cannot reduce the time-dependent Schro¨dinger equation to an eigenvalue equation. However, if the time-dependent terms can be regarded as small perturbations, we can develop a form of perturbation theory that can be used to calculate the effects of time-dependent terms. Consider a system for which the Hamiltonian may be written as the sum of a dominant unperturbed part Hˆ0 and a small perturbation Hˆ ′ as Hˆ = Hˆ0 + λHˆ ′(t) , (10.1.1) where the unperturbed Hamiltonian Hˆ0 is independent of time, whereas the perturbation Hˆ ′ may depend on time. The parameter 0 < λ < 1 has been introduced to keep track of various orders of approximations as in the case of time-independent perturbation theory discussed in Chapter 8. The basic idea is that a small perturbation to the Hamiltonian will produce a small change in the wave function. For a time-dependent perturbation, this change means that a system, initially in a particular eigenstate of Hˆ0, will find itself in a time-dependent admixture of the eigenstates. Thus the time-dependent perturbation causes the system to undergo transition to different eigenstates of Hˆ0. So for time-dependent perturbations our interest lies in calculating the probabilities for transition to different states.