Von Neumann equations with time-dependent Hamiltonians and supersymmetric quantum mechanics

Abstract

Starting with a time-independent Hamiltonian $h$ and an appropriately chosen solution of the von Neumann equation $i\dot{\rho }(t)=[h,\rho (t)]$ we construct its binary-Darboux partner $h_1(t)$ and an exact scattering solution of $i{\dot{\rho }}_1(t)=[h_1(t),{\rho }_1(t)]$ where $h_1(t)$ is time-dependent and not isospectral to $h$. The method is analogous to supersymmetric quantum mechanics but is based on a different version of a Darboux transformation. We illustrate the technique by the example where $h$ corresponds to a 1-D harmonic oscillator. The resulting $h_1(t)$ represents a scattering of a soliton-like pulse on a three-level system.

revtex, 3 eps files

References in corpus (3)

• Darboux-integrable nonlinear Liouville-von Neumann equation
• Microscopic Foundation of Nonextensive Statistics
• Nonlinear von Neumann-type equations: Darboux invariance and spectra

Cited by in corpus (4)

• Quantum morphogenesis: A variation on Thom's catastrophe theory
• Darboux-integration of idρ/dt=[H,f(ρ)]
• Special solutions of nonlinear von Neumann equations
• Solving the von Neumann equation with time-dependent Hamiltonian. Part I: Method