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Journal of Jinggangshan University

UNDERSTANDING THE QUANTUM ADIABATIC THEOREM IN THE REPRESENTATION OF THE INSTANTANEOUS EIGENSTATE

Xiaobing Luo

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

Using representation transformation, we transform an exatly solvable two-level Hamiltonian into a time-independent one in the representation of the instantaneous eigenstate.In terms of the basis of the instantaneous eigenstate, the physical picture of the quantum adiabatic theorem becomes very clear, which will help us understand the quantum adiabatic theorem.

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Using representation transformation, we transform an exatly solvable two-level Hamiltonian into a time-independent one in the representation of the instantaneous eigenstate.In terms of the basis of the instantaneous eigenstate, the physical picture of the quantum adiabatic theorem becomes very clear, which will help us understand the quantum adiabatic theorem.
We construct the adiabatic theorem independent of the quantum representations. The quantum-representation-independent geometrical phase, adiabatic condition, and fidelity of the adiabatic approximation are derived. As an example, we apply the quantum-representation-independent adiabatic theorem to provide a thorough resolution of the Marzlin-Sanders inconsistency. We show that the Marzlin-Sanders inconsistency results from the wrong identification of the time-dependent Hamiltonian as the instantaneous energy operator and the quantum representation dependence of the Marzlin-Sanders transformation.
The adiabatic theorem states that an initial eigenstate of a slowly varying Hamiltonian remains close to an instantaneous eigenstate of the Hamiltonian at a later time. We show that a perfunctory application of this statement is problematic if the change in eigenstate is significant, regardless of how closely the evolution satisfies the requirements of the adiabatic theorem. We also introduce an example of a two-level system with an exactly solvable evolution to demonstrate the inapplicability of the adiabatic approximation for a particular slowly varying Hamiltonian.
A new and intuitive perturbative approach to time-dependent quantum mechanics problems is presented, which is useful in situations where the evolution of the Hamiltonian is slow. The state of a system which starts in an instantaneous eigenstate of the initial Hamiltonian is written as a power series which has a straightforward diagrammatic representation. Each term of the series corresponds to a sequence of ``adiabatic'' evolutions, during which the system remains in an instantaneous eigenstate of the Hamiltonian, punctuated by transitions from one state to another. The first term of this series is the standard adiabatic evolution, the next is the well known first correction to it, and subsequent terms can be written down essentially by inspection. Although the final result is perhaps not terribly surprising, it seems to not be widely known, and the interpretation is new, as far as we know. Application of the method to the adiabatic approximation is given, and some discussion of the validity of this approximation is presented.
Abstract. A general quantum adiabatic theorem with and without the time-dependent orthogonalization is proven, which can be applied to understand the origin of activation energies in chemical reactions. Further proofs are also developed for the oscillating Schwinger Hamiltonian to establish the relation-ship between the internal (due to time-dependent eigenfunctions) and external (due to time-dependent Hamiltonian) timescales. We prove that this relation-ship needs to be taken as an independent condition for the quantum adiabatic approximation. We give four examples, including logical expositions based on the spin- 1 2 two-level system to address the gapped and gapless (due to energy level crossings) systems, as well as to understand how does this theorem allows one to study dynamical systems such as chemical reactions.
Transition amplitudes between instantaneous eigenstates of a quantum two-level system are evaluated analytically on the basis of a new parametrization of its evolution operator, which has recently been proposed to construct exact solutions. In particular, the condition under which the transitions are suppressed is examined analytically. It is shown that the analytic expression of the transition amplitude enables us, not only to confirm the adiabatic theorem, but also to derive the necessary and sufficient condition for quantum two-level system to remain in one of the instantaneous eigenstates.
A general quantum adiabatic theorem with and without the time-dependent orthogonalization is proven, which can be applied to understand the origin of activation energies in chemical reactions. Further proofs are also developed for the oscillating Schwinger Hamiltonian to establish the relationship between the internal (due to time-dependent eigenfunctions) and external (due to time-dependent Hamiltonian) timescales. We prove that this relationship needs to be taken as an independent condition for the quantum adiabatic approximation. We give four examples, including logical expositions based on the spin-½ two-level system to address the gapped and gapless (due to energy level crossings) systems, as well as to understand how does this theorem allows one to study dynamical systems such as chemical reactions.