Exactly solvable models with time-dependent boundary conditions

N. N. Bogolyubov, A. G. Shumovskaya, A. S. Shumovskiǐ

Ashok V. Pimpale, Mohsen Razavy

Quantum tunneling in one spatial dimension in the presence of time-dependent potentials is investigated theoretically. First, a general multichannel formulation of the problem is given for harmonic time-dependence. The case of an oscillatory delta-function potential of constant strength is discussed at length and specific numerical calculations are presented for the reflection and transmission coefficients. Then other exactly solvable time-dependent potentials are obtained by carrying out successive unitary transformations. As an example of this class, a delta-function potential with time-dependent strength and boundary conditions is studied. Finally tunneling time-delay for time-dependent potentials is formulated in the multichannel situation, and also the concept of first passage time for the decay of a wave packet confined to one well of a double well potential is generalized for the case of time-dependent barrier and boundary conditions.

Ranjan Kumar Ghosh, Sumathi Rao

Adam J. Makowski

It is shown here that the non-stationary Schrodinger equation can be solved exactly for two quantum models subject to Dirichlet boundary conditions. One of them is a modified problem of a quantum bouncer, i.e. the problem of a particle falling down in the gravitational field on a moving (oscillating) platform such as a loudspeaker. The second model is a 'cuff-off oscillator' with a moving infinite potential wall and a time-dependent frequency. In both cases exact solutions are given in closed forms, easy to use. Their possible applications are also indicated. In each of the models extra coordinate- and time-dependent phase factors are generated by moving boundaries in the former case giving rise to a non-local effect in quantum mechanics.

Jiang Yi

Quantum impurity models have deep connection with the boundary problem.Low-dimensional materials have been an additional source of motivation to investigate onedimensional exactly solvable models.We will study the common Hubbard model,and then study the Hubbard model with open boundary conditions.

Bernard Derrida, Michel Mendès France, Jacques Peyri�re