Physical Review A

Gauge-invariant periodic solutions of the exact time-dependent Schrödinger equation and their time averages

Kit-Keung Kan, James J. Griffin, T. W. Atwater, Maria Dworzecka

Published Jan 1, 1983• 14 citations

Abstract

The gauge-invariant periodic quantization (GIPQ) method is applied within the complete Hilbert space of solutions of the exact time-dependent Schr\"odinger equation. The multiplicity of such GIPQ solutions is shown to exceed by far the number of eigenfunctions they are intended to analogize. However, time averaging (or the equivalent averaging over all values of the progressive phase) reduces in this space to a projection upon the exact eigenfunctions under which every one of these GIPQ solutions either vanishes identically or reduces precisely to one of the stationary (but unnormalized) eigenfunctions of the exact Hamiltonian. One concludes that the correspondence between the GIPQ solutions and exact eigenfunctions is well manageable in the complete Hilbert space, a result which encourages the further study of time-averaged GIPQ solutions in parametric subspaces, where time averaging projects not upon an eigenstate, but upon the constant component of the GIPQ periodic solution.

DOI: 10.1103/physreva.27.12

Related papers

Gauge invariant periodic quantization method

Kit-Keung Kan

Based on the variational principle in a general manifold in the Hilbert space and the principle of gauge invariance, the general gauge invariant periodic quantization method is formulated. This method, which is a direct generalization of the one previously obtained in the case of time-dependent Hartree-Fock, leads to a quantization rule which takes the Bohr-Sommerfeld form when the variational parameters are transformed to canonical variables. The method is illustrated by applying it to the manifold of minimum wave packets for a simple harmonic oscillator. It leads to the exact energy spectrum and a discrete set of wave packets which have maximum overlap with the exact eigenstates. The method is also applied to the exact Schr\"odinger dynamics where it exposes some open questions of the method.[NUCLEAR STRUCTURE Time-dependent description of many-body bound states. Gauge invariant periodic quantization method, time-dependent variational principle and their canonical formulation.]

Physical Review CJul 1, 198120 citations

Exact time dependence of solutions to the time-dependent Schrödinger equation

Max A. Lohe

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.

Journal of Physics A Mathematical and TheoreticalDec 9, 200865 citations

Quantum canonical transformations and exact solution of the Schrödinger equation

Ali Mostafazadeh

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/9611052Journal of Mathematical PhysicsJul 1, 199717 citations

Gauge-invariant periodic quantized good-parity states in Lipkin-Meshkov-Glick model

Kit-Keung Kan

The gauge-invariant periodic quantization method is applied to time-dependent states with definite parities in the Lipkin-Meshkov-Glick model. It is shown that the resulting discrete energy spectrum is free of discontinuity and predicts the correct number of states for all interaction strengths. It offers therefore a qualitative improvement over the gauge-invariant periodic quantized time-dependent Hartree-Fock states of this model.NUCLEAR STRUCTURE Time-dependent description of many-body bound states, gauge-invariant periodic quantization method, Lipkin-Meshkov-Glick model, good-parity time-dependent states.

Physical Review CNov 1, 198010 citations

On the Quantization of One-Dimensional Nonstationary Coulomb Potential System

Salah Menouar, Mustapha Maamache, Jeong Ryeol Choi +1 more

Exact solutions of the one-dimensional Schrödinger equation with a time-dependent Coulomb potential [- z ( t )/| x |] are investigated using the invariant method (Lewis and Riesenfeld theorem) together with unitary transformation approach. The eigenfunctions and the corresponding eigenvalues of the system are obtained analytically. When the time dependence of all coefficients vanishes, our results exactly reduce to those known for stationary case.

Journal of the Physical Society of JapanMay 18, 20122 citations

Geometric Phases, Symmetries of Dynamical Invariants, and Exact Solution of the Schrödinger Equation

We introduce the notion of the geometrically equivalent quantum systems (GEQS) as quantum systems that lead to the same geometric phases for a given complete set of initial state vectors. We give a characterization of the GEQS. These systems have a common dynamical invariant, and their Hamiltonians and evolution operators are related by symmetry transformations of the invariant. If the invariant is T-periodic, the corresponding class of GEQS includes a system with a T-periodic Hamiltonian. We apply our general results to study the classes of GEQS that include a system with a cranked Hamiltonian H(t) = e −iKt H0e iKt. We show that the cranking operator K also belongs to this class. Hence, in spite of the fact that it is time-independent, it leads to nontrivial cyclic evolutions and geometric phases. Our analysis allows for an explicit construction of a complete set of nonstationary cyclic states of any time-independent simple harmonic oscillator. The period of these cyclic states is half the characteristic period of the oscillator. PACS numbers: 03.65.Bz 1

Jan 1, 20010 citations