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WKB wave functions for one dimensional nonrelativistic problems by simple transformation derivation, solving by application of Liouville substitution to Schroedinger equation
Numerical integration of Schroedinger equation for spontaneous ionization of hydrogen atom in electric field
The inverse scattering problem is introduced in the context of the Schroedinger equation. The emphasis is on the Gelfand Levitan approach to the nonrelativistic problem at fixed angular momentum.
Methods of solution of the inverse scattering problem at fixed energy in quantum mechanics are presented. Scattering experiments of a beam of particles at a nonrelativisitic energy by a target made up of particles are analyzed. The Schroedinger equation is used to develop the quantum mechanical description of the system and one of several functions depending on the relative distance of the particles. The inverse problem is the construction of the potentials from experimental measurements.
The solution of the one dimensional, time independent, Schroedinger equation in which the energy minus the potential varies as the nth power of the distance is obtained from proper linear combinations of Bessel functions. The linear combinations called generalized Airy functions, reduce to the usual Airy functions Ai(x) and Bi(x) when n equals 1 and have the same type of simple asymptotic behavior. Expressions for the generalized Airy functions which can be evaluated by the method of generalized Gaussian quadrature are obtained.
A rapid numerical technique for the determination of bound states of a partial-wave-projected Schroedinger equation is presented. First, one needs to integrate the equation only outwards as in the scattering case, and second, the number of trials necessary to determine the eigenenergy and the corresponding eigenfunction is considerably less than in the usual method. As a nontrivial example of the technique, bound states are calculated in the exchange approximation for the e-/He+ system and l equals 1 partial wave.
Using a space translation operation, it is shown that the Schroedinger equation for an atom in an external electromagnetic field can be solved with sufficient accuracy to obtain the transition probabilities for multiphoton absorption from a monochromatic laser beam of arbitrary intensity or frequency. The specific applicability conditions of the method are discussed.
Improved values of the proton impact excitation cross sections at coronal energies for all the Fe(+12) ground configuration transitions are presented. These were obtained by direct computer integration of the Schroedinger equation (with the states expressed in intermediate coupling) resulting from the semiclassical Coulomb excitation theory formulation of the process. Comparison is made with previous results. The associated rate constants at coronal temperatures are given and compared with the corresponding electron impact excitation rate constants.
The application of the basic formalistic elements of wave mechanics theory is discussed. The theory is used to describe the physical phenomena on the microscopic level, the fluid dynamics of gases and liquids, and the analysis of physical phenomena on the macroscopic (visually observable) level. The practical advantages of relating the two fields of wave mechanics and fluid mechanics through the use of the Schroedinger equation constitute the approach to this relationship. Some of the subjects include: (1) fundamental aspects of wave mechanics theory, (2) laminarity of flow, (3) velocity potential, (4) disturbances in fluids, (5) introductory elements of the bifurcation theory, and (6) physiological aspects in fluid dynamics.
The method of multiple scales is used to derive a nonlinear Schroedinger equation for the temporal and spatial modulation of the amplitudes and the phases of waves propagating in a hard-walled circular duct. This equation is used to show that monochromatic waves are stable and to determine the amplitude dependance of the cutoff frequencies.
Exact numerical calculations are made for scattering of quantum mechanical particles hitting a square two-dimensional potential barrier (an exact analog of the Goos-Haenchen optical experiments). Quantum mechanical streamlines are plotted and found to be smooth and continuous, to have continuous first derivatives even through the classical forbidden region, and to form quantized vortices around each of the nodal points. A comparison is made between the present numerical calculations and the stationary wave approximation, and good agreement is found between both the Goos-Haenchen shifts and the reflection coefficients. The time-independent Schroedinger equation for real wavefunctions is reduced to solving a nonlinear first-order partial differential equation, leading to a generalization of the Prager-Hirschfelder perturbation scheme. Implications of the hydrodynamical formulation of quantum mechanics are discussed, and cases are cited where quantum and classical mechanical motions are identical.
A semiclassical model proposed by Pearson and Hansen (1974) for computing collision-induced transition probabilities in diatomic molecules is tested by the direct-simulation Monte Carlo method. Specifically, this model is described by point centers of repulsion for collision dynamics, and the resulting classical trajectories are used in conjunction with the Schroedinger equation for a rigid-rotator harmonic oscillator to compute the rotational energy transition probabilities necessary to evaluate the rotation-translation exchange phenomena. It is assumed that a single, average energy spacing exists between the initial state and possible final states for a given collision.
The method of multiple scales is used to derive a nonlinear Schroedinger equation for the temporal and spatial modulation of the amplitudes and the phases of waves propagating in a hard-walled circular duct. This equation is used to show that monochromatic waves are stable and to determine the amplitude dependance of the cut off frequencies.
The two-particle, steady-state Schroedinger equation is transformed to center of mass and internuclear distance vector coordinates, leading to the free particle wave equation for the kinetic energy motion of the molecule and a decoupled wave equation for a single particle of reduced mass moving in a spherical potential field. The latter describes the vibrational and rotational energy modes of the diatomic molecule. For fixed internuclear distance, this becomes the equation of rigid rotator motion. The classical partition function for the rotator is derived and compared with the quantum expression. Molecular symmetry effects are developed from the generalized Pauli principle that the steady-state wave function of any system of fundamental particles must be antisymmetric. Nuclear spin and spin quantum functions are introduced and ortho- and para-states of rotators, along with their degeneracies, are defined. Effects of nuclear spin on entropy are deduced. Next, rigid polyatomic rotators are considered and the partition function for this case is derived. The patterns of rotational energy levels for nonlinear molecules are discussed for the spherical symmetric top, for the prolate symmetric top, for the oblate symmetric top, and for the asymmetric top. Finally, the equilibrium energy and specific heat of rigid rotators are derived.
Values of proton collisional cross-sections at coronal energies have been computer for all transitions between magnetic sublevels of the Fe(+12) and Fe(+13) ground configurations. Semiclassical Coulomb excitation theory was used with the resulting truncated Schroedinger equations being directly integrated. The corresponding rate constants for coronal temperatures are presented and comparison is made with previous results.
The absolute intensity of the 4-0 vibration band of CO is measured in spectra obtained using a 25-m base-path multiple-traversal absorption cell and a 5-m scanning spectrometer. The intensities of individual vibration-rotation lines in this band are determined from measurements of their equivalent widths, and absolute values for the rotationless transition moment and the vibration-rotation interaction factor are derived from the measured line strengths. The experimentally obtained vibration-rotation function is compared with a theoretical curve; agreement between theory and experiment is found to be good for the P-branch but poor for the R-branch. It is noted that numerical solutions to the radial Schroedinger equation lead to vibration-rotation function values that are in good agreement with the experiment.
It is shown through the use of transformation theory that unique semiclassical atomic scattering states which obey the asymptotic conditions of formal scattering theory can be derived by transforming 'nontraveling' atomic states, ie., states whose coordinate variables are referred to a stationary origin, to frames at rest with respect to the incoming or outgoing particles. An overview of the problem of properly defining such scattering states is presented. The operator which carries out the necessary transformation from inertial to noninertial frames is derived and its properties are discussed. The relation of this transformation operator to the 'translation factor' discussed in the literature is presented. The application of this operator to transform the time-dependent Schroedinger equation from an inertial to a noninertial frame is presented and shown to introduce new terms in the resulting equation. The implications of these new terms to scattering problems are discussed.
Proton collisional excitation cross sections at solar coronal energies for the Fe(11+) /3S(2)3p(3)/ ground-configuration transitions are presented. Semiclassical Coulomb excitation theory is used with direct integration of the resulting Schroedinger equation. The rate constants at coronal temperatures are compared with the corresponding ones for electron impact excitation.