Showing papers on "Finite potential well published in 1989"

Variational boundary perturbation theory for enclosed quantum systems

Abstract: The authors propose a new method of solving the Schrodinger equation for a quantum system enclosed in a box with infinite potential walls. The method combines the variational technique with boundary perturbation theory and can be applied to a general non-separable case. When the Schrodinger equation for an enclosed system separates into an ordinary differential equation, the method gives the exact energy and wavefunction. As an application of the method the authors obtain the ground-state energy of the hydrogen atom placed off-centre in a spherical cavity. A generalisation of the variational boundary perturbation technique for finite potential walls is suggested.

Multiple scattering approach to one‐dimensional potential problems

Abstract: By calculating the reflection and transmission coefficients for a square potential barrier in terms of a series of scatterings at step potential discontinuities, a multiple scattering approach to potential problems is presented. This approach is applied to WKB potential problems by interpreting the WKB connection formulas in terms of reflection and transmission coefficients at each turning point. An algorithm for the systematic construction of the energy Green’s function is given. The significance of complex coordinate turning points is demonstrated. The equivalence of this approach and that of the Gaussian approximation of the path integral is mentioned briefly.

Square-well gas collision integrals and diffusion in a square-well Lorentz gas

Abstract: A representation for collision cross sections of a square‐well dilute gas mixture is developed that reduces their computation to the evaluation of one‐dimensional integrals with integrands of elementary closed form. Consequently, the computation of collision integrals and Chapman–Enskog and Kihara transport‐coefficient approximations, which depend on the collision cross sections, is analytically and numerically simplified. These results are then applied to the limiting case of a Lorentz gas, i.e., a dilute gas of mobile particles diffusing through a bed of scatterers that can be regarded as fixed. In particular, our results are used to evaluate the exact diffusion coefficient DL of a Lorentz gas with square‐well potential interactions as a function of both square‐well width and temperature, i.e., square‐well depth. (We treat here only the case in which the scatterers are dilute enough for the problem to be accurately described by a Boltzmann equation.) The exact DL is compared with its first and second Ch...

Interesting features of relativistic and nonrelativistic quantal bound states in one, two, and three dimensions

Abstract: Various aspects of critically bound quantal ground states close to the threshold in one, two, and three dimensions are examined in nonrelativistic quantum mechanics using attractive square‐well and delta‐function potentials. The mathematical and physical reasons are presented for the nonoccurrence of the bound state in the case of three dimensions and its occurrence in one and two dimensions for infinitesimally small strengths and ranges of the potential. Also analyzed is the condition for critical binding in relativistic quantum mechanics using the Klein–Gordon equation in one, two, and three dimensions. Corresponding features for the case of the Dirac equation in one and three dimensions are briefly discussed.

The equivalent two-body and delta energy lower bounds for N-boson systems

Abstract: Studies the ground-state energy of a system of N identical bosons, each having mass m, which interact in one dimension via the pair potential V(x)= gamma f(x/a) and obey non-relativistic quantum mechanics. A recent energy lower bound based on the known exact solution for the delta potential is compared to an earlier bound provided by the equivalent two-body method. The delta bound is good whenever the potential is very narrow and deep. For any bounded potential, the earlier energy bound is always better for large N. Detailed results and graphs are given for the sech-squared potential and the square-well potential.

Transition‐state theory for quantum and classical particle escape from a finite square well

Abstract: A precise statement of transition‐state theory is given. The theory is made intuitive and precise by the introduction of a ‘‘semipermeable’’ barrier between reactant and product. The theory is then illustrated by calculating, both quantum and classical mechanically, the rate of particle escape from a finite square well. The two calculations are fundamentally different and illuminate different aspects of the theory. Finally, the solution is recast into the Arrhenian form with which it is usually identified, and physically satisfying expressions for the rate prefactor, activation energy, and activation entropy are obtained.

Fluids in the Gravitational Field Exact Treatment of One Dimensional Systems

Abstract: Partition functions and thermodynamic functions for fluids in one-dimensional model systems interacting with a gravitational field are investigated. The intermolecular potential is chosen in a manner that only next-neighbour particles interact, which allows for an exact evaluation of the partition function using elementary methods. The local contributions to the thermodynamic functions are determined by the local pressure. The variation of the Gibbs potential and the density as functions of the height, may show a point of inflexion as real two phase systems. This is discussed for rigid particles interacting with a square well potential, and allowed to assume two states of different lengths.

Computer simulation of depolarized light scattering from diatoms with hard core and square well interactions at low temperatures

Abstract: We show that the depolarized light scattered from two interacting atoms is qualitatively different for a purely repulsive hard core potential and partially attractive square well potential. This difference is due to the bound states possible with attractive potentials and is used to obtain the spectra from these dimers. The spectra were calculated from the autocorrelation of the scattered electric field found by an effective field model. Values were phase space averaged using Gaussian integration. Molecular dynamics routines evolved the phase space points in time for the autocorrelation. The hard core spectra showed an exponential decay characterized by the mean passage time. The square well spectra consisted of a hard core‐type term, plus a harmonic component due to dimer contributions. Subtraction of the hard core spectra from the square well spectra gave the dimer component. The dimer component of the spectra was proportional to the square of the number of dimers and was characterized by the thermal ro...