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arXiv:quant-ph/9904026 · Journal of Mathematical Physics

Generalized adiabatic product expansion: A nonperturbative method of solving the time-dependent Schrödinger equation

Ali Mostafazadeh

Published Jul 1, 1999• 12 citations
arXivHTMLPDF

Abstract

We outline a method based on successive canonical transformations which yields a product expansion for the evolution operator of a general (possibly non-Hermitian) Hamiltonian. For a class of such Hamiltonians this expansion involves a finite number of terms, and our method gives the exact solution of the corresponding time-dependent Schrödinger equation. We apply this method to study the dynamics of a general nondegenerate two-level quantum system, a time-dependent classical harmonic oscillator, and a degenerate system consisting of a spin 1 particle interacting with a time-dependent electric field E⃗(t) through the Stark Hamiltonian H=λ(J⃗⋅E⃗)2.

DOI
10.1063/1.532889

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Time-dependent unitary transformations are used to study the Schrödinger equation for explicitly time-dependent Hamiltonians of the form H(t)=R(t)⋅J, where R is an arbitrary real vector-valued function of time and J is the angular momentum operator. The solution of the Schrödinger equation for the most general Hamiltonian of this form is shown to be equivalent to the special case R=(1,0,ν(t)). This corresponds to the problem of a driven two-level atom for the spin half representation of J. It is also shown that by requiring the magnitude of R to depend on its direction in a particular way, one can solve the Schrödinger equation exactly. In particular, it is shown that for every Hamiltonian of the form H(t)=R(t)⋅J there is another Hamiltonian with the same eigenstates for which the Schrödinger equation is exactly solved. The application of the results to the exact solution of the parallel transport equation and exact holomony calculation for SU(2) principal bundles (Yang–Mills gauge theory) is also pointed out.

arXiv:quant-ph/9611052•Journal of Mathematical Physics•Jul 1, 1997•17 citations
We outline a method based on successive canonical transformations which yields a product expansion for the evolution operator of a general (possibly non-Hermitian) Hamiltonian. For a class of such Hamiltonians this expansion involves a finite number of terms, and our method gives the exact solution of the corresponding time-dependent Schrödinger equation. We apply this method to study the dynamics of a general nondegenerate two-level quantum system, a time-dependent classical harmonic oscillator, and a degenerate system consisting of a spin 1 particle interacting with a time-dependent electric field E⃗(t) through the Stark Hamiltonian H=λ(J⃗⋅E⃗)2.
The time-evolution operator for an explicitly time-dependent Hamiltonian is expressed as the product of a sequence of unitary operators. These are obtained by successive time-dependent unitary transformations of the Hilbert space followed by the adiabatic approximation at each step. The resulting adiabatic product expansion yields a generalization of the quantum adiabatic approximation. Furthermore, it leads to an infinite class of exactly solvable models. PACS number: 03.65.Bz Consider the dynamics of a quantum mechanical system whose Hamiltonian H = H(τ) is explicitly time-dependent. The evolution of a state vector |ψ(τ) 〉 is governed by the Schrödinger equation: i¯h | ˙ ψ(τ) 〉 = H(τ) |ψ(τ) 〉 , |ψ(0) 〉 = |ψ0 〉 , (1) where a dot means a time-derivative. Alternatively, one has |ψ(τ) 〉 = U(τ)|ψ0〉, where i τ − H(t) dt U(τ) = T e ¯h 0 (2) is the time-evolution operator. Here T denotes the time-ordering operator. The purpose of this article is to express U(τ) as the product of a sequence of unitary operators each of
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.
The quantum-mechanical version of the four kinds of classical canonical transformations is investigated by using non-Hermitian operator techniques. To help understand the usefulness of this approach, the eigenvalue problem of a harmonic oscillator is solved in two different types of canonical transformations. The quantum form of the classical Hamilton-Jacobi theory is also employed to solve time-dependent Schr\"odinger wave equations, showing that when one uses the classical action as a generating function of the quantum canonical transformation of time evolutions of state vectors, the corresponding propagator can easily be obtained.
A precise definition of an adiabaticity parameter \ensuremath{\nu} of a time-dependent Hamiltonian is proposed. A variation of the time-dependent perturbation theory is presented which yields a series expansion of the evolution operator U(\ensuremath{\tau})=${\ensuremath{\sum}}_{\mathrm{\ensuremath{\ell}}}$${\mathrm{U}}^{(\mathrm{\ensuremath{\ell}})}$(\ensuremath{\tau}) with ${\mathrm{U}}^{(\mathrm{\ensuremath{\ell}})}$(\ensuremath{\tau}) being at least of the order ${\ensuremath{\nu}}^{\mathrm{\ensuremath{\ell}}}$. In particular, U(0){\mathrm{U}}^{(0)}U(0)(\ensuremath{\tau}) corresponds to the adiabatic approximation and yields Berry's adiabatic phase. It is shown that this series expansion has nothing to do with the 1/\ensuremath{\tau} expansion of U(\ensuremath{\tau}). It is also shown that the nonadiabatic part of the evolution operator is generated by a transformed Hamiltonian which is off-diagonal in the eigenbasis of the initial Hamiltonian. This suggests the introduction of an adiabatic product expansion for U(\ensuremath{\tau}) which turns out to yield exact expressions for U(\ensuremath{\tau}) for a large number of quantum systems. In particular, a simple application of the adiabatic product expansion is used to show that for the Hamiltonian describing the dynamics of a magnetic dipole in an arbitrarily changing magnetic field, there exists another Hamiltonian with the same eigenvectors for which the Schr\"odinger equation is exactly solvable. Some related issues concerning geometric phases and their physical significance are also discussed.
Solutions of the Schrödinger equation with an exact time dependence are derived as eigenfunctions of dynamical invariants which are constructed from time-independent operators using time-dependent unitary transformations. Exact solutions and a closed form expression for the corresponding time evolution operator are found for a wide range of time-dependent Hamiltonians in d dimensions, including non-Hermitean -symmetric Hamiltonians. Hamiltonians are constructed using time-dependent unitary spatial transformations comprising dilatations, translations and rotations and solutions are found in several forms: as eigenfunctions of a quadratic invariant, as coherent state eigenfunctions of boson operators, as plane wave solutions from which the general solution is obtained as an integral transform by means of the Fourier transform, and as distributional solutions for which the initial wavefunction is the Dirac δ-function. For the isotropic harmonic oscillator in d dimensions radial solutions are found which extend known results for d = 1, including Barut–Girardello and Perelomov coherent states (i.e., vector coherent states), which are shown to be related to eigenfunctions of the quadratic invariant by the ζ-transformation. This transformation, which leaves the Ermakov equation invariant, implements SU(1, 1) transformations on linear dynamical invariants. coherent states are derived also for the time-dependent linear potential. Exact solutions are found for Hamiltonians with electromagnetic interactions in which the time-dependent magnetic and electric fields are not necessarily spatially uniform. As an example, it is shown how to find exact solutions of the time-dependent Schrödinger equation for the Dirac magnetic monopole in the presence of time-dependent magnetic and electric fields of a specified form.
Time-dependent unitary transformations are used to study the Schrödinger equation for explicitly time-dependent Hamiltonians of the form H(t)=R(t)⋅J, where R is an arbitrary real vector-valued function of time and J is the angular momentum operator. The solution of the Schrödinger equation for the most general Hamiltonian of this form is shown to be equivalent to the special case R=(1,0,ν(t)). This corresponds to the problem of a driven two-level atom for the spin half representation of J. It is also shown that by requiring the magnitude of R to depend on its direction in a particular way, one can solve the Schrödinger equation exactly. In particular, it is shown that for every Hamiltonian of the form H(t)=R(t)⋅J there is another Hamiltonian with the same eigenstates for which the Schrödinger equation is exactly solved. The application of the results to the exact solution of the parallel transport equation and exact holomony calculation for SU(2) principal bundles (Yang–Mills gauge theory) is also pointed out.