A new method previously elaborated on in time-dependent quantum mechanics is recalled and used to establish, on the basis of the unperturbed Hamiltonian, explicit and tractable expressions of the long-term density matrix, well suited to transport problems. A constant collision potential and a constant external field, switched on at t=0, are assumed. As an illustration, the general results are applied to the quantum theory of electrical conductivity in the case of quasifree electrons.
\nthe quantum calculation of the d.c. electrical conductivity for an electron system is reconsidered by using an improved form of the long-time solution of the density matrix. The starting time-evolution equation which is written as a linear system in the Liouville space of discrete quantum states, is first solved in terms of Cramer's determinants and next suitably adapted to tranport problems. The calculation of the statistical average of the many-body current density, in the presence of scattering potential, is carried out in the thermodynamic limit, using a characteristic property of collision matrices introduced earlier by van Hove as the so-called
delta-singularity. In contrast to previous methods based on kinetic equations solved by perturbations, Kubo's theory, or Mori type memory functions, the resolution of an integral equation, initially included in Cramer's formulae, is no longer required. Among its major advantages, the method succeeds in overcoming the well known zero frequency divergent behaviour of the conductivity. It also yields an explicit expression of an effective relaxation time which is just constructed out of those diagonal transition sequences responsible for self-energy effects in van Hove's theory. This expression which only assumes the randomization of phases of initial states can be used to get tractable expansions in powers of the collision potential. Its range of validity is limited by the effect of collisions on equilibrium statistics, which is shown to be negligible within the Peierls criterion,
hbar/taullmu,
tau being the relaxation time and
mu the Fermi energy. Finally, the comparison of the present approach with the elementary transport theory reveals that the relaxation time is close to the standard expression, even though the latter would be exactly recovered, in lowest order of collisions, only on disregarding divergent terms arising beyond second order.\n
The determinantal formalism previously built up in the evolution operator problem is extended to the density matrix equation in the case, here, of a Heaviside step external perturbation, and in the presence of internal collision potential. Reduced determinantal forms, matched to those first derived for the evolution operator, are obtained and compared with both the iterative expansion and the initial Fredholm-Laplace solution. They not only exhibit the multiple transition structure of the physical response, along with the associated transition width upon expansion, but also the trace conservation requirement is shown to be satisfied at any order and all times. An illustration is given in a simplified physical system.
For pt.I see ibid. vol.16, p.3987-4000, (1983). The determinantal formalism for solving the time-dependent Schrodinger equation of the density matrix has been developed, in I, assuming a constant external perturbation. The important case of an harmonic perturbation is presented here. Detailed expressions are derived for diagonal and nondiagonal elements of the response, and the trace conservation theorem is shown to be satisfied at all times and any order. The results are again applied to the simple illustrative quantum system considered in I.
For Lindblad's master equation of open quantum systems with a general quadratic form of the Hamiltonian, the propagator of the density matrix is analytically calculated by using path integral techniques. The time-dependent density matrix is applied to nuclear barrier penetration in heavy ion collisions with inverted oscillator and double-well potentials. The quantum mechanical decoherence of pairs of phase space histories in the propagator is studied and shown that the decoherence depends crucially on the transport coefficients.
Forms of canonical (Bloch) and Dirac density matrices for free electrons with and without external applied fields are first considered. The basic tool employed is the Bloch equation with a one-electron Hamiltonian. Such an approach is used to obtain a perturbation theory to all orders for the idempotent Dirac density matrix when a common potential energy V(r) is switched on to originally free electrons. The relation to density functional theory is then considered and the exchange–correlation contribution Vxc(r) to V(r) is expressed in terms of first- and second-order density matrices following Holas and March. These latter density matrices are now for the fully interacting system and, in particular, the first-order density matrix is no longer idempotent, though it must still satisfy generalized Pauli Principle conditions. Reference is also made to a localized Wigner electron in a strong magnetic field.
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.