Exact Quantum Mechanical Solution Research Articles

Multiphonon absorption in ionic crystals

Calculations of the frequency and temperature dependence of infrared multiphonon absorption for the alkali halides and alkaline-earth flourides are compared with previously published data and some new results. The temperature dependence of the infrared absorption of pure crystals of NaF, NaCl, and KCl at 10.6 \ensuremath{\mu}m from room temperature to close to the melting point are reported. The calculations are based upon a simple approach involving and interatomic Morse-potential function in which lattice dispersion is introduced via a multiphonon frequency distribution. Good agreement between experiment and theory is obtained in which the lattice density-of-states and thermal-expansion-coefficient data (which determine the degree of anharmonicity) are the principal input to the calculations. Some structure in the multiphonon spectrum is predicted for some compounds at low temperatures, which has not yet been observed expermentally. Since the Morse potential has an exact quantum-mechanical solution, the anharmonicity is contained in the calculations without resorting to perturbation theory.

Accuracy in determining the potential energy curve minimum

The symbol “ r e ” commonly refers to two quantities: 1. (1) the location of the minumum of the actual rotationless potential energy function; 2. (2) the value calculated from spectroscopic data using the equation r e = ( h ̷ 4πμcB e ) 1 2 Are the two quantities equivalent? To investigate this question, the exact quantum mechanical solutions of the Davidson potential, U(r) = ar 2 + b r 2 − 2(ab) 1 2 , are analyzed analytically in order to determine the r e of definition 2. Comparison is then possible with the r e of definition 1. Disagreement is found between the two quantities called r e for several of the spectroscopic analysis methods tested. These disagreements, generalized to diatomic potentials, indicate possible errors in the determination of the actual minimum up to 0.1% with three of the tested analysis methods and less than 10 −5% with other, more appropriate methods.

Quantum mechanical close coupling approach to molecular collisions. jz -conserving coupled states approximation

New coupled equations describing collisions of an atom and a diatomic molecule are derived in this paper. By utilizing a description of the collision in terms of rotating coordinates, all coupling in the z component of angular momentum is isolated into purely kinematic effects. By neglecting these couplings, one is led to approximate equations for which the jz component of angular momentum for the molecule is conserved. In addition, the scattering cross sections are formulated by neglecting the effect on the wavefunction of the rotation of the coordinate axes so that in place of Wigner rotation matrices dmmJ (Θ) appearing, one deals with simple Legendre polynomials and the orbital angular momentum l2 is approximated by l(l + 1) ℏ2. It is noted that the procedure involves no approximations so far as the potential matrix elements are concerned. Furthermore, the number of equations remaining coupled is drastically reduced and a completely quantum mechanical description of the dynamics of both internal states and relative motion is retained. The physical implications of the approximations are examined, and it is seen that the neglect of intermultiplet coupling gives rise to consideration of only transitions where both the orientation and magnitude of the rotor angular momentum change. Further, the neglect of transformation effects on the wavefunction is expected to be least accurate for the inelastic forward scattering and best for backward scattering and the j =0→0 elastic scattering. Finally, the present simplest version of the approximation obviously is not intended for treating processes dependent on mj transitions, e.g., NMR relaxation in He–H2. Next the formalism is applied in test calculations to He–H2 collisions using the Krauss-Mies potential energy surface. Numerical results for elastic and inelastic integral and differential cross sections are compared with exact quantum mechanical close coupling solutions of the standard coupled channel equations. Over the energy range studied (from 0.1 eV up to 0.9 eV), agreement to within a few percent is obtained. Additional coupled states calculations are reported at 1.2 eV and computation times are compared against those required for a full close coupling solution. Calculations for the Roberts He–H2 surface are also reported to illustrate the independence of the approximations on the strength of the coupling (so long as the inelastic scattering is predominantly in the backward direction). The dramatic savings afforded by the present approach are such as to make possible fully converged calculations at collision energies typically studied in molecular beam experiments. Thus, for elastic and inelastic nonreactive collisions, involving a repulsive-type interaction, the approach makes the a priori quantum mechanical description of the scattering of a diatom by an atom practical.

Linear adiabatic invariants and coherent states

The Born-Fock adiabatic theorem is extended to all orders for some quadratic quantum systems with finitely or infinitely degenerate energy spectra. A prescription is given for obtaining adiabatic invariants to any order. For any quadratic quantum system with N degrees of freedom there are 2N linear adiabatic invariant series, which correspond to the 2N exact invariants. The exact quantum mechanical solution for any nonstationary quadratic quantum system is also constructed by making use of the coherent-state representation: The Green's function, coherent states, transition amplitudes and probabilities and their generating functions are obtained explicitly. Two particular systems, the N-dimensional time-dependent general oscillator and charged particle motion in a varying and uniform electromagnetic field, are considered in greater detail as examples.

Low temperature properties of one-dimensional metals for the interrupted strand model

The low temperature specific heat and the frequency-dependent polarizability α ω are calculated for the interrupted strand model of Kuse and Zeller. The average specific heat is shown to be exponentially attenuated. An exact quantum mechanical solution at T = 0 of α ω displays an interesting low-frequency behavior, reflecting the finite level spacing at the Fermi energy.

Exact Quantum-Mechanical Calculation of Vibrational Energy Transfer to an Oscillator by Collision with a Particle

In a recent paper,! Secrest and Johnson (SJ) pre­sented an exact quantum-mechanical solution for vibrational transition probabilities in the collinear collision of a particle with a harmonic oscillator. Their approach utilized an elegant Green's function procedure in terms of plane waves. It was found that their results did not reduce to the first-order distorted-wave ap­proximation (FODWA) of Jackson and Mott

Free paths and transport phenomena gases and the quantum theory of collisions. I.—The rigid sphere model

The necessity for the use of quantum mechanics in the theory of atomic phenomena is most clearly manifest in the study of collision processes. Diffrac­tion effects have been observed in the scattering of electrons from crystals and by atoms, while the recent developments of molecular ray technique have made it possible to establish the existence of cross-grating spectra in the reflection of molecular beams from crystal surfaces. In view of the importance of wave theory in these phenomena, it is clearly necessary to examine the conditions under which the classical theory of gases must be modified and to determine the nature of the modifications. Such an investigation receives added importance owing to the possibility of experimental test by molecular ray methods. Also, considerable interest is attached to the possibility of direct experimental proof of the Bose-Einstein statistics for neutral atoms and molecules from collision experiments as has already been possible for α-particles. In order to develop the quantum theory of collisions in a form suitable for this purpose, we first discuss the simplest model which bears sufficient re­semblance to the actual facts, and so we consider the rigid sphere model for gas atoms. This model has already proved valuable in the classical theory of transport phenomena and has the additional advantage of permitting an exact quantum mechanical solution. It will be seen that the results obtained by the use of this model are of great interest and suggest several new lines of investigation, both experimental and theoretical. Finally, a method for dealing with the general case of any law of force will be discussed.