Parasupersymmetry of the one-dimensional time-dependent Schrödinger equation is established. It is intimately connected with a chain of the time-dependent Darboux transformations. As an example a parasupersymmetric model of nonrelativistic free particle with threefold degenerate discrete spectrum of an integral of motion is constructed.
For Dirac equation, operator-invariants containing explicit time-dependence in parallel to known time-dependent invariants of nonrelativistic Schrödinger equation are introduced and discussed. As an example, a free Dirac particle is considered and new invariants are constructed for it. The integral of motion, which is initial Newton–Wigner position operator, is obtained explicitly for a free Dirac particle. For such particle with kick modeled by delta-function of time, the time-depending integral, which has physical meaning of initial momentum, is found. 1
Starting with a time-independent Hamiltonian h and an appropriately chosen solution of the von Neumann equation $i\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\rho}}(t)=[h,\ensuremath{\rho}(t)]$ we construct its binary-Darboux partner h1(t) and an exact scattering solution of $i{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\rho}}}_{1}(t)=[{h}_{1}(t),{\ensuremath{\rho}}_{1}(t)],$ where h1(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 one-dimensional harmonic oscillator. The resulting h1(t) represents a scattering of a solitonlike pulse on a three-level system.
Abstract This chapter the reviews essential concepts and methods of time-dependent quantum mechanics. (1) Formal solutions of the time-dependent Schrödinger equation. (2) The two-level system. (3) Interacting systems: The time-dependent self-consistent field approximation and the Ehrenfest approximation. (4) The Born–Oppenheimer approximation and nuclear potential surfaces. (5) Time-dependent Hamiltonians: A two-level system in a time-dependent field. (6) Non-adiabatic dynamics: The surface-hopping method. (7) Expressing the time evolution in terms of the Green operator. (8) Representations (the Schrödinger and Heisenberg representations, the interaction representation). (9) Time-dependent perturbation theory. (10) Quantum dynamics of the free particles. (11) Quantum dynamics of the harmonic oscillator. (12) The raising/lowering operators formalism. (13) Tunneling.