Classical and quantum mechanics of the damped harmonic oscillator

Arrival time in quantum mechanics

Seismic migration and inversion describe a class of closely related processes sharing common objectives and underlying physical principles. These processes range in complexity from the simple NMO‐stack to the complex, iterative, multidimensional, prestack, nonlinear inversion used in the elastic seismic case. By making use of amplitudes versus offset, it is, in principle, possible to determine the three elastic parameters from compressional data. NMO‐stack can be modified to solve for these parameters, as can prestack migration. Linearized, wave‐equation inversion does not inordinately increase the complexity of data processing. The principal part of a migration‐inversion algorithm is the migration. Practical difficulties are considerable, including both correctable and intrinsic limitations in data quality, limitations in current algorithms (which we hope are correctable), and correctable (or perhaps intrinsic) limitations in computer power.

Migration and inversion of seismic data

On the time operator in quantum mechanics and the heisenberg uncertainty relation for energy and time

http://streaming.ictp.it/preprints/P/01/067.pdf

Unified time analysis of photon and particle tunnelling

Conjugate functions of various classical Hamiltonians are studied and different methods for their construction are presented. From these classical functions, for many dynamical systems, Hermitian operators can be found which depend linearly on time and are conjugates of Hamiltonian operators. The eigenvalues of conjugate operators, under certain conditions, represent the result of measurement of time in nonrelativistic quantum mechanics. Because of this property « time operators » may be used to study the quantum theory of microscopic clocks and to determine their accuracy. For noninteracting relativistic particles with spin zero and spin one-half a direct method of obtaining time operator is given. These conjugate operators have been used to derive well-known relations of the time delay and the rate of change of phase shift with energy in scattering theory.

Quantum-mechanical conjugate of the hamiltonian operator

For a central potential the Schrodinger equation for the lth partial wave is first transformed to a differential equation which depends linearly on l, and then the latter form is approximated by a difference equation. The structure of this difference equation is such that it makes it possible to arrive, in a natural way, at a Thiele's interpolation formula for the logarithmic derivative of the wave function outside the potential, which is related to the discrete form of the -matrix. If the numerical values of the -matrix, at a given energy, are known for N partial waves, then the found interpolation formula can be used to determine the potential at N equidistant points. This method, in a sense, complements the Newton–Sabatier method since it works better for low energy scattering. It can easily be generalized and applied to the problems where the potential depends on angular momentum, the example of inverse problem for spin-orbit interaction has been studied as an application of this method. Numerical res...

A discrete method of constructing central and angular-momentum dependent potentials

The Schrodinger equation for the sum of a central and a noncentral potential is solved approximately by wirting the wave function in terms of two variable quantities, a real phase and a complex amplitude. The variable phase satisfies the Hamilton-Jacobi equation which contains the central part of the potential only and, therefore, the phase is expressible as a definite integral. The complex amplitude, on the other hand, satisfies a nonlinear partial differential equation in which the noncentral part of the potential appears. This amplitude is calculated approximately first by using the technique of quasi-linearization and then by solving the resulting linear equation by the method of characteristics. In this way the asymptotic phase and the amplitude of the wave function are obtained from the recursive solution of a set of linear differential equations. The method has been applied to find the scattering phase shifts for several separable and nonseparable potential functions.

A semi-classical approximation for two-dimensional scattering by nonseparable potentials

If the acoustic wave velocity and the density of an inhomogeneous medium are dependent on one of the coordinates only, then it is possible to express the logarithmic derivative of the pressure along this coordinate as a nonlinear first‐order differential equation. In the present work a particular finite difference approximation to this equation is found which allows one to obtain the reflection coefficient as a ratio of two polynomials involving the sine of the angle of incidence. In this way one can find, in a very simple way, the unknown coefficients of the two polynomials from the values of the reflection coefficient for different angles, and at the same time one can determine the wave velocity and the density, from the unique continued fraction expansion of this rational representation for two different frequencies.

A method for constructing wave velocity and density profiles from the angular dependence of the reflection coefficient

A many-body system consisting of a central elastically bound particle coupled to a large number of harmonic oscillators is studied. The co-ordinate of this particle satisfies the equation of motion of a damped harmonic oscillator. To get the quantal description of this damped motion, the many-body wave function for the complete system is written in the coherent-state basis,i.e. as a product of a real stationary wave function and a co-ordinate- and time-dependent phase factor. In order to isolate the motion of the central particle, the stationary wave function is approximated by a product of single-particle wave functions. By calculating the contribution of the variable phase factor to the single-particle wave function, it is shown that the resulting equation for the damped motion of an oscillator is the Schrodinger-Langevin equation.

M. Razavy

Papers

Quantum-mechanical conjugate of the hamiltonian operator

A discrete method of constructing central and angular-momentum dependent potentials

A semi-classical approximation for two-dimensional scattering by nonseparable potentials

A method for constructing wave velocity and density profiles from the angular dependence of the reflection coefficient

Coherent states and the irreversible motion of a particle coupled to a system of oscillators