Exact Wave Functions of Time-Dependent Hamiltonian Systems Involving Quadratic, Inverse Quadratic, and $(1/\hat x)\hat p + \hat p(1/\hat x)$ TermsExact Wave Functions of Time-Dependent Hamiltonian Systems Involving Quadratic, Inverse Quadratic, and $$(1/\hat x)\hat p + \hat p(1/\hat x)$$ Terms

Jeong Ryeol Choi

International Journal of Theoretical PhysicsApr 1, 200317 citations

Inverse problem of quadratic time-dependent Hamiltonians

Guangjie Guo, Yan Meng, Hong Chang +2 more

Using an algebraic approach, it is possible to obtain the temporal evolution wave function for a Gaussian wave-packet obeying the quadratic time-dependent Hamiltonian (QTDH). However, in general, most of the practical cases are not exactly solvable, for we need general solutions of the Riccatti equations which are not generally known. We therefore bypass directly solving for the temporal evolution wave function, and study its inverse problem. We start with a particular evolution of the wave-packet, and get the required Hamiltonian by using the inverse method. The inverse approach opens up a new way to find new exact solutions to the QTDH. Some typical examples are studied in detail. For a specific time-dependent periodic harmonic oscillator, the Berry phase is obtained exactly.Using an algebraic approach, it is possible to obtain the temporal evolution wave function for a Gaussian wave-packet obeying the quadratic time-dependent Hamiltonian (QTDH). However, in general, most of the practical cases are not exactly solvable, for we need general solutions of the Riccatti equations which are not generally known. We therefore bypass directly solving for the temporal evolution wave function, and study its inverse problem. We start with a particular evolution of the wave-packet, and get the required Hamiltonian by using the inverse method. The inverse approach opens up a new way to find new exact solutions to the QTDH. Some typical examples are studied in detail. For a specific time-dependent periodic harmonic oscillator, the Berry phase is obtained exactly.

Chinese Physics BAug 1, 20150 citations

Unitarily manipulating in time and space a Gaussian wave-packet motional state of a single atom in a quadratic potential field

Xijia Miao

The paper first discusses theoretically the off-resonance selective excitation method that is dependent on the atomic internal states and used to generate approximately a standard coherent state of harmonic oscillator. The coherent average method then is proposed to construct the state-selective trigger pulse. A state-selective trigger pulse can keep Gaussian shape unchanged but change in an internal-state-dependent form the center-of-mass position and/or momentum of an atomic Gaussian wave-packet motional state. A Gaussian wave-packet state is one of the simplest wave-packet states that can be easily manipulated and controlled in time and space. The paper also investigates how to manipulate in time and space an atomic Gaussian wave-packet motional state by a generalized quadratic potential field. A general quadratic Hamiltonian can affect not only the center-of-mass position and momentum but also the complex linewidth of a Gaussian wave-packet motional state while keep Gaussian shape of the motional state unchanged. It is shown that generally quadratic terms of a quadratic Hamiltonian can control directly the complex linewidth, while linear terms of a quadratic Hamiltonian can affect only the center-of-mass position and momentum of a Gaussian wave-packet motional state.The paper first discusses theoretically the off-resonance selective excitation method that is dependent on the atomic internal states and used to generate approximately a standard coherent state of harmonic oscillator. The coherent average method then is proposed to construct the state-selective trigger pulse. A state-selective trigger pulse can keep Gaussian shape unchanged but change in an internal-state-dependent form the center-of-mass position and/or momentum of an atomic Gaussian wave-packet motional state. A Gaussian wave-packet state is one of the simplest wave-packet states that can be easily manipulated and controlled in time and space. The paper also investigates how to manipulate in time and space an atomic Gaussian wave-packet motional state by a generalized quadratic potential field. A general quadratic Hamiltonian can affect not only the center-of-mass position and momentum but also the complex linewidth of a Gaussian wave-packet motional state while keep Gaussian shape of the motional state unchanged. It is shown that generally quadratic terms of a quadratic Hamiltonian can control directly the complex linewidth, while linear terms of a quadratic Hamiltonian can affect only the center-of-mass position and momentum of a Gaussian wave-packet motional state.

arXiv:0708.2129Aug 16, 20070 citations

Time-Dependent Schrödinger Equation and Gaussian Wave Packets

Dieter Schuch

Fundamental theories of physicsJan 1, 20180 citations

Hamiltonian approach for wave packet dynamics: Beyond Gaussian wave functions

Andrey Pereverzev, Eric R. Bittner

arXiv:0806.4170Physics Letters AMay 8, 20093 citations

Invariant operators for quadratic Hamiltonians

Mark Andrews

The motion of wave packets can be easily determined for any Hamiltonian that is quadratic in position and momentum, even if the coefficients of the terms in the Hamiltonian vary with time. The method is based on the existence of an invariant operator, linear in position and momentum, the coefficients of these operators being solutions of the corresponding classical system. This immediately yields a set of very simple wave packets whose evolution is easily determined, and whose magnitude has the same form as the energy eigenfunctions of the harmonic oscillator (i.e., Gaussian or Hermite–Gaussian). This set provides a complete basis for finding the evolution of any state and the exact propagator is readily determined. For the harmonic oscillator, this set includes coherent states, squeezed states, displaced number states, and squeezed number states as special cases. The following important properties hold for every quadratic Hamiltonian, no matter what time dependence is present in the Hamiltonian: (1) The motion of the centroid of any wave packet separates from that of any moments relative to the centroid. (2) Every detail of the evolution of the quantum system can be calculated from the solutions of the corresponding classical system. (3) Wave functions of Gaussian (or Hermite–Gaussian) form will retain that form as they evolve.The motion of wave packets can be easily determined for any Hamiltonian that is quadratic in position and momentum, even if the coefficients of the terms in the Hamiltonian vary with time. The method is based on the existence of an invariant operator, linear in position and momentum, the coefficients of these operators being solutions of the corresponding classical system. This immediately yields a set of very simple wave packets whose evolution is easily determined, and whose magnitude has the same form as the energy eigenfunctions of the harmonic oscillator (i.e., Gaussian or Hermite–Gaussian). This set provides a complete basis for finding the evolution of any state and the exact propagator is readily determined. For the harmonic oscillator, this set includes coherent states, squeezed states, displaced number states, and squeezed number states as special cases. The following important properties hold for every quadratic Hamiltonian, no matter what time dependence is present in the Hamiltonian: (1) The motion of the centroid of any wave packet separates from that of any moments relative to the centroid. (2) Every detail of the evolution of the quantum system can be calculated from the solutions of the corresponding classical system. (3) Wave functions of Gaussian (or Hermite–Gaussian) form will retain that form as they evolve.

American Journal of PhysicsApr 1, 199923 citations

Exact Wave Functions of Time-Dependent Hamiltonian Systems Involving Quadratic, Inverse Quadratic, and

(1/\hat x)\hat p + \hat p(1/\hat x)

Terms