A recursive solution of Heisenberg's equation and its interpretation

Abstract

We present the generalization of the recursion method of Haydock and co-workers to systems of many interacting particles. This new method has close similarities to the memory function or Mori formalism, but with some important differences. Heisenberg's equation for the time evolution of a microscopic operator is recursively transformed into a tridiagonal matrix equation. This equation resolves the operator into components corresponding to transitions of different energies. The projected spectrum of transitions has a continued fraction expansion given by the elements of the tridiagonal matrix. We show that for an appropriate choice of inner product this density of transitions obeys a generalization of the black body theorem of electromagnetism, in that it is exponentially insensitive to distant parts of the system. This implies that the projected density of transitions is computationally stable and can be calculated even in macroscopic many-body systems. We argue that the physical content of the density of transitions is determined by the nature of its singular points, such as discrete transitions, continuous spectrum, band edges and van Hove singularities.

Convergent method for calculating the properties of many interacting electrons

“…An efficient and accurate way of calculating the projected resolvent is with a continued fraction expansion [4] R u (E) = b 0 2 / (E -a 0 -b 1 2 / (E -a 1 -... -b n 2 / (E -a n -...) ...)),…”

“…While the stationary states of interacting systems have energies which grow with the size of the system, the stationary transitions, operators transforming one stationary state into another, need not have energies which depend on the system's size because they are the differences between energies of stationary states. The transitions satisfy Heisenberg's equation [4] which is equivalent to Schroedinger's equation, but does not become singular as the system gets larger, again because the transition energies do not increase. Just as non-interacting states are projected on a localized orbital for the PDoS, stationary transitions are projected on a localized operator for the PDoT which is then the total density of transitions weighted by the probability that each transition is induced by the localized operator on which it is being projected.…”

Closed Quantum Black-Scholes: Quantum Drift and the Heisenberg Equation of Motion

“…We start in section 2, by deriving the dynamics for quantized financial observables using the Heisenberg equation of motion. We apply a geometric technique to deriving eigenfunctions, before discussing how recursive techniques (for example see [2]) can be used to generate power series solutions to the closed quantum Black-Scholes. This technique involves introducing a Riemannian metric in order to simplify the integration, and functional form for the eigenfunctions.…”

Closed Quantum Black-Scholes: Quantum Drift and the Heisenberg Equation of Motion

“…(2.28) Under this coordinate system our Hamiltonian becomes (we use a slight abuse of notation, by still writing the independent variable as x): A formal solution to this equation can be written (see [2]):…”