Some remarks about the time-dependent Schrödinger equation with damping

Abstract

The missing derivation of the time-dependent Schrödinger equation following Schrödinger's original description of the time-independent Schrödinger equation. Also, this description is extended to derive the Caldirola-Kanai, the Schuch-Schrödinger, and the Gisin-Schrödinger equation. In the second part, the Gisin-Schrödinger equation will be derived once more using the Ito formalism of stochastic differential equations. Furthermore, we discuss the extension to larger spin-system using the cluster mean-field theory.

Investigation of the magnetic phase diagram of the 2D ferrimagnetic J 1–J 2 model

“…Spins of different clusters are not entangled. Within this publication, the time-dependence is taken into account by solving the modified Gisin-Schrödinger equation [34][35][36][37]. Therefore, the t-CMFT can calculate the ferrimagnetic J 1 -J 2 model's ground state configuration and its dynamics.…”

Investigation of the temperature phase diagram of the 2D semi-classical ferrimagnetic J1−J2 model

“…In general, most spin states are entangled. Exceptions are the ferromagnetic state and coherent spin states [46][47][48][49][50]. These quantum states are not entangled.…”

“…In order to get from the regular Heisenberg model with entanglement to the semi-classical description without entanglement, we replace all spin operators ⃗ S 2 with which the considered spin ⃗ S 1 interacts with the corresponding spin expectation value ⟨ ⃗ S 2 ⟩ = ⟨ψ| ⃗ S 2 |ψ⟩. Here |ψ⟩ is either the complete quantum state of the system (product state of coherent spin states) or a local part (a single coherent spin state at the corresponding lattice site) [40,49,50].…”

“…This fact means that the eigenstates of Ĥ are coherent spin states. The dynamics of our system is given the modified Gisin-Schrödinger equation [45,[50][51][52]. However, this partial differential equation describes the dynamics at zero temperature T = 0.…”

“…Our aim, however, is to describe it at T > 0. We first clarify that the modified Gisin-Schrödinger equation can be seen as the quantummechanical analog of the classical Landau-Lifshitz-Gilbert (LLG) equation [53][54][55] in the case of a single spin, respectively, coherent spin states [50]. However, the LLG equation also describes spin dynamics at T = 0.…”