Dae-Yup Song

Tunneling and energy splitting in an asymmetric double-well potential

An asymmetric double-well potential is considered, assuming that the minima of the wells are quadratic with a frequency x and the difference of the minima is close to a multiple ofx. A WKB wave function is constructed on both sides of the local maximum between the wells, by matching the WKB function to the exact wave functions near the classical turning points. The continuities of the wave function and its first derivative at the local maximum then give the energy-level splitting formula, which not only repro- duces the instanton result for a symmetric potential, but also elu- cidates the appearance of resonances of tunneling in the asymmetric potential.

Geometric phase, Hannay's angle, and an exact action variable

Canonical structure of a generalized time-periodic harmonic oscillator is studied by finding the exact action variable (invariant). Hannays angle is defined if closed curves of constant action variables return to the same curves in phase space after a time evolution. The condition for the existence of Hannays angle turns out to be identical to that for the existence of a complete set of (quasi)periodic wave functions. Hannays angle is calculated, and it is shown that Berrys relation of semiclassical origin on geometric phase and Hannays angle is exact for the cases considered.

Unitary relation between a harmonic oscillator of time-dependent frequency and a simple harmonic oscillator with and without an inverse-square potential

The unitary operator which transforms a harmonic oscillator system of time-dependent frequency into that of a simple harmonic oscillator of different time-scale is found, with and without an inverse-square potential. It is shown that for both cases, this operator can be used in finding complete sets of wave functions of a generalized harmonic oscillator system from the well-known sets of the simple harmonic oscillator. Exact invariants of the time-dependent systems can also be obtained from the constant Hamiltonians of unit mass and frequency by making use of this unitary transformation. The geometric phases for the wave functions of a generalized harmonic oscillator with an inverse-square potential are given.

Generalization of the Darboux transformation and generalized harmonic oscillators

The Darbroux transformation is generalized for time-dependent Hamiltonian systems which include a term linear in momentum and a time-dependent mass. The formalism for the N-fold application of the transformation is also established, and these formalisms are applied for a general quadratic system (a generalized harmonic oscillator) and a quadratic system with an inverse-square interaction up to N = 2. Among the new features found, it is shown, for the general quadratic system, that the shape of potential difference between the original system and the transformed system could oscillate according to a classical solution, which is related to the existence of coherent states in the system.

Periodic Hamiltonian and berry's phase in harmonic oscillators

For a time-dependent

Unitary transformation for the system of a particle in a linear potential

\tau

Unitary relations in time-dependent harmonic oscillators

-periodic harmonic oscillator of two linearly independent homogeneous solutions of classical equation of motion which are bounded all over the time (stable), it is shown, there is a representation of states cyclic up to multiplicative constants under

Localization or tunneling in asymmetric double-well potentials

\tau

Exact coherent states of a noninteracting Fermi gas in a harmonic trap

-evolution or

Collective motions of a quantum gas confined in a harmonic trap

2\tau