Exact solution of coupled Schrodinger equations in a stationary n-state exponential model (10 pages)

Diwaker, Bandhan Panda, Aniruddha Chakraborty

Max Teubner

The $n$-state exponential model refers to coupled Schr\"odinger equations with potential ${V}_{ij}(x)={U}_{i}{\ensuremath{\delta}}_{ij}+{V}_{ij}{e}^{\ensuremath{-}\ensuremath{\alpha}x}$. Exact solutions in terms of Meijer's $G$ functions are obtained, provided the ${V}_{ij}$ factorize according to ${V}_{ij}=\ifmmode\pm\else\textpm\fi{}{V}_{i}{V}_{j}$. The nonadiabatic transition matrix $N$ is determined.

Vladimir I. Osherov, Hiroki Nakamura

Quantum mechanically exact analytical solutions are obtained for a two-state exponential model, in which the exponent of diabatic coupling is one-half of that of the diabatic potential curve. A very simple and accurate semiclassical formula is found for the nonadiabatic transition probability. This gives a direct generalization of the Landau-Zener and Rosen-Zener formulas.

Margarit Rizea

Ognyan Oreshkov, Denis Bouvy

We challenge the common belief that if there is no absolute time parameter in physics, a quantum system described without reference to an external clock can be assumed to be in a stationary state of its Hamiltonian. We present a time-reparameterisation invariant quantum evolution law, which for a given initial condition predicts the same trajectory in state space as the Schrödinger equation, except that for nontrivial trajectories it does not predict the speed at which the trajectory is traversed. The solutions of this evolution law are all time-reparameterised solutions of the Schrödinger equation. We show how the predictions of the Schrödinger equation are recovered relative to an internal clock in this framework. In contrast to the Page-Wootters formalism or Dirac's quantisation of the Hamiltonian constraint, here the global sate is not stationary. We discuss the assumptions leading to the common conclusion that the state can be taken stationary and suggest that they need to be revisited.

A. P. Machil'skii