We consider exactly solvable models in quantum mechanics with potential-energy functions that exhibit exponential dependence on the coordinate. As particular examples we choose a spherically symmetric well to illustrate the concept of critical strengths, and a barrier in one dimension to discuss tunnelling.
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.
We show that the list of analytically solvable potentials in nonrelativistic quantum mechanics can be considerably enlarged. In particular, we show that those noncentral potentials for which the Schrödinger equation is separable are analytically solvable provided the separated problem for each of the coordinates belongs to the class of exactly solvable one dimensional problems. As an illustration, we discuss in detail two examples, one in two and the other in three dimensions. A list of analytically solvable noncentral potentials in spherical polar coordinates is also given. Extension of these ideas to other standard orthogonal coordinate systems as well as to higher dimensions is straightforward.
In the paper the description of infinite movement in the functional formulation of classical mechanics is investigated. On the example of simple exactly solvable problems (passing through the barrier and falling in the center) the two classes of problems of scattering and singularity are considered. The functional mechanics corrections, arising from scattering, to the mean values and variance of canonical variables are calculated. In particular in the simplest case of transmission through the barrier the shift of the mean value coordinate by a constant arises , this constant depends on the parameters of the barrier, and logarithmic correction to the variance of the free motion coordinate. Also it is shown, that functional mechanics approach leads to the elimination of singularities in the kinetic energy of the falling in the center, which is equivalent to the solution of the Friedman equation in cosmology.