arXiv subjects
Publications and source records attributed to Zafar Ahmed.
The summation of the partial wave series for Coulomb scattering amplitude, $f^C(θ)$ is avoided because the series is oscillatorily and divergent. Instead, $f^C(θ)$ is obtained by solving the Schr{ö}dinger equation in parabolic cylindrical co-ordinates which is not a general method. Here, we show that a reconstructed series, $(1-\cosθ) ^2f^C(θ)$, is both convergent and analytically summable.
Employing the concept of time-delay, a relation is found which counts the number of quantal resonances supported by a potential. Several simple and advanced illustrations include a treatment of square-well, Dirac delta barrier, an interesting physical situation from neutron reflectometry, and the Delta resonance appearing in the scattering of πmeson from proton.
We present the case of quantal transmission through a smooth, single-piece exponential potential, $V(x)=-V_0 e^{x/a}$, in contrast to the piece-wise continuous potentials, as a pedagogic model to demonstrate the analytic extraction of transmission (reflection) co-efficient.
We define pseudo-reality and pseudo-adjointness of a Hamiltonian, $H$, as $ρH ρ^{-1}=H^\ast$ and $μH μ^{-1}=H^\prime$, respectively. We prove that the former yields the {\it necessary} condition for spectrum to be real whereas the latter helps in fixing a definition for inner-product of the eigenstates. Here we separate out adjointness of an operator from its Hermitian-adjointness. It turns out that a Hamiltonian possessing real spectrum is first pseudo-real, further it could be Hermitian, PT-symmetric or pseudo-Hermitian.
Currently, it has been claimed that certain Hermitian Hamiltonians have parity (P) and they are PT-invariant. We propose generalized definitions of time-reversal operator (T) and orthonormality such that all Hermitian Hamiltonians are P, T, PT, and CPT invariant. The PT-norm and CPT-norm are indefinite and definite respectively. The energy-eigenstates are either E-type (e.g., even) or O-type (e.g., odd). C mimics the charge-conjugation symmetry which is recently found to exist for a non-Hermitian Hamiltonian. For a Hermitian Hamiltonian it coincides with P.
We propose construction of a unique and definite metric ($η_+$), time-reversal operator (T) and an inner product such that the pseudo-Hermitian matrix Hamiltonians are C, PT, and CPT invariant and PT(CPT)-norm is indefinite (definite). Here, P and C denote the generalized symmetries : parity and charge-conjugation respectively. The limitations of the other current approaches have been brought out.
We study almost-forward scattering in the context of usual and non-commutative QED. We study the semi-classical behaviour of particles undergoing this scattering process in the two theories, and show that the shock wave picture, effective in QED fails for NCQED. Further, we show that whereas in QED, there are no leading logarithmic contributions to the amplitude upto sixth order, uncancelled logarithms appear in NCQED.
Employing the currently discussed notion of pseudo-Hermiticity, we define a pseudo-unitary group. Further, we develop a random matrix theory which is invariant under such a group and call this ensemble of pseudo-Hermitian random matrices as the pseudo-unitary ensemble. We obtain exact results for the nearest-neighbour level spacing distribution for (2 X 2) PT-symmetric Hamiltonian matrices which has a novel form, s log (1/s) near zero spacing. This shows a level repulsion in marked distinction with an algebraic form in the Wigner surmise. We believe that this paves way for a description of varied phenomena in two-dimensional statistical mechanics, quantum chromodynamics, and so on.
We propose that the real spectrum and the orthogonality of the states for several known complex potentials of both types, PT-symmetric and non-PT-symmetric can be understood in terms of currently proposed $η$-pseudo-Hermiticity (Mostafazadeh, quant-ph/0107001) of a Hamiltonian, provided the Hermitian linear automorphism, $η$, is introduced as $e^{-θp}$ which affects an imaginary shift of the co-ordinate : $e^{-θp} x e^{θp} = x+iθ$.
We present a modified Brownian motion model for random matrices where the eigenvalues (or levels) of a random matrix evolve in "time" in such a way that they never cross each other's path. Also, owing to the exact integrability of the level dynamics, we incorporate long-time recurrences into the random walk problem underlying the Brownian motion. From this model, we derive the Coulomb interaction between the two eigenvalues. We further show that the Coulomb gas analogy fails if the confining potential, $V(E)$ is a transcendental function such that there exist orthogonal polynomials with weighting function, $\exp [-βE]$, where $β$ is a symmetry parameter.