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arXiv:1312.7504

Non-adiabatic transition probability with a moving $δ$ potential coupling

Diwaker, Aniruddha Chakraborty

Published Dec 29, 2013• 0 citations
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Abstract

The present work focuses on the calculation of a non-adiabatic transition probability between two states which may or may not cross with each other and are coupled to each other by a moving $δ$ function potential. Here, the time dependent Schrodinger equation is converted to time independent one by using a scaling factor which is function of time. This time independent Schrodinger equation is then considered for two potentials coupled by a moving $δ$ potential and an expression for non-adiabatic transition probability has been derived.

DOI
10.48550/arxiv.1312.7504

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Non-adiabatic transition probability with a moving δδδ potential coupling
The present work focuses on the calculation of a non-adiabatic transition probability between two states which may or may not cross with each other and are coupled to each other by a moving δδδ function potential. Here, the time dependent Schrodinger equation is converted to time independent one by using a scaling factor which is function of time. This time independent Schrodinger equation is then considered for two potentials coupled by a moving δδδ potential and an expression for non-adiabatic transition probability has been derived.
We obtain probabilities at a crossing of two linearly time-dependent potentials that are constantly coupled to the other by solving a time-dependent Schrödinger equation. We find that the system which was initially localized at one state evolves to split into both states at the crossing. The probability splitting depends on the coupling strength V0V_0V0​ such that the system stays at the initial state in its entirety when V0V_0V0​ = 0 while it is divided equally in both states when V0→∞V_0 \rightarrow {\infty}V0​→∞ . For a finite coupling the probability branching at the crossing is not even and thus a complete probability transfer at t→∞t \rightarrow {\infty}t→∞ is not achieved in the linear potential crossing problem. The Landau-Zener formula for transition probability at t→∞t \rightarrow {\infty}t→∞ is expressed in terms of the probabilities at the crossing.
We have proposed an analytical method for finding the exact solution for the two-state scattering problems, where two flat potentials are coupled by a rectangular potential. We found a way to express the two-state problem by an effective single-state problem, where it is easy to estimate the transition probability from one potential to the other by estimating the amount of missing flux from the first potential. This has been an unsolved problem for a long time and is of considerable importance as a model for non-adiabatic electronic transition.
The time-dependent Schrödinger equation with a time-dependent delta function potential is solved exactly for many special cases. In all other cases the problem can be reduced to an integral equation of the Volterra type. It is shown that by knowing the wavefunction at the origin, one may derive the wavefunction everywhere. Thus, the problem is reduced from a PDE in two variables to an integral equation in one. These results are used to compare adiabatic versus sudden changes in the potential. It is shown that adiabatic changes in the potential lead to the conservation of the normalization of the probability density.
The nonstationary Schr\"odinger equation is considered in a finite basis of states. The model Hamiltonian matrix corresponds to a single diabatic potential curve with a Coulombic $\ensuremath{\sim}1/t$ time dependence. An arbitrary number of other diabatic potential curves are flat, i.e., time independent and have arbitrary energies. Related states are coupled by constant interactions with the Coulomb state. The resulting nonstationary Schr\"odinger equation is solved by the method of contour integral. Probabilities of transitions to any other state are obtained as $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty}$ in a simple analytical form for the case when the Coulomb state is populated initially (at instant of time $\stackrel{\ensuremath{\rightarrow}}{t}+0).$ The formulas apply both to the cases when a horizontal diabatic potential curve is crossed by the Coulomb one and to a noncrossing situation. In the limit of weak coupling, the transition probabilities are interpreted in terms of a sequence of pairwise Landau-Zener-type transitions. Mapping of the Coulomb model onto an exactly solvable exponential multistate model is established. For the special two-state case, the well-known Nikitin model is recovered.
Non-adiabatic transitions in two-level systems are investigated theoretically for a random time dependence of ħω, the energy difference, between the levels. We assumed that ω = ω(x) and the coordinate x = x(t) is a random function of time. Diffusion and Poisson processes (both homogeneous and with a source) for x(t) were assumed. The cases of linear crossing terms (ω = γx) and non-linear terms (ω = ω e exp (- αx) + ω0) were considered. Values of the non-adiabatic transition probability per unit time were obtained by perturbation theory for ω1τ c ≪ 1 where τ c is the correlation time and ω1 is the off-diagonal matrix element.