Searcharxiv⌕ Search

arXiv subjects

Zafar Ahmed

Publications and source records attributed to Zafar Ahmed.

At least 19 recordsLinked to original sources
arXiv2025

New Measurements of the Deuteron to Proton F2 Structure Function Ratio

Nucleon structure functions, as measured in lepton-nucleon scattering, have historically provided a critical observable in the study of partonic dynamics within the nucleon. However, at very large parton momenta it is both experimentally and theoretically challenging to extract parton distributions due to the probable onset of non-perturbative contributions and the unavailability of high precision data at critical kinematics. Extraction of the neutron structure and the d-quark distribution have been further challenging due to the necessity of applying nuclear corrections when utilizing scattering data from a deuteron target to extract free neutron structure. However, a program of experiments has been carried out recently at the energy-upgraded Jefferson Lab electron accelerator aimed at significantly reducing the nuclear correction uncertainties on the d-quark distribution function at large partonic momentum. This allows leveraging the vast body of deuterium data covering a large kinematic range to be utilized for d-quark parton distribution function extraction. We present new data from experiment E12-10-002 carried out in Jefferson Lab Hall C on the deuteron to proton cross-section ratio at large BJorken-x. These results significantly improve the precision of existing data, and provide a first look at the expected impact on quark distributions extracted from global parton distribution function fits.

hep-ex↗
arXiv2021

Bound state eigenvalues from transmission coefficients

Experts know that bound state energy eigenvalues of a potential well are poles of its transmission amplitude, $t(E)$. The textbooks do well by solving the bound states and scattering states separately, but the connection goes unheeded. Here, we present interesting and instructive illustrations, where we extract bound state eigenvalues from the known transmission coefficients $(T(E)=|t(E)|^2)$ for six analytically solvable potential wells.

quant-ph↗
arXiv2021

Scattering states and bound states of exponential potentials

We explore the relationships between scattering states and bound states of different non-analytic segments (depending on $|x|$) of the exponential potential, and elucidate the status of the special scattering states found in an earlier publication by Ahmed et al. A similar analysis can be made of non-analytic segments of power potentials such as $x^3$.

quant-ph↗
arXiv2020

An update on coherent scattering from complex non-PT-symmetric Scarf II potential with new analytic forms

The versatile and exactly solvable Scarf II has been predicting, confirming and demonstrating interesting phenomena in complex PT-symmetric sector, most impressively. However, for the non-PT-symmetric sector it has gone underutilized. Here, we present most simple analytic forms for the scattering coefficients $(T(k),R(k),|\det S(k)|)$. On one hand, these forms demonstrate earlier effects and confirm the recent ones. On the other hand they make new predictions - all simply and analytically. We show the possibilities of both self-dual and non-self-dual spectral singularities (NSDSS) in two non-PT sectors (potentials). The former one is not accompanied by time-reversed coherent perfect absorption (CPA) and gives rise to the parametrically controlled splitting of SS in to a finite number of complex conjugate pairs of eigenvalues (CCPEs). The latter ones (NSDSS) behave just oppositely: CPA but no splitting of SS. We demonstrate a one-sided reflectionlessness without invisibility. Most importantly, we bring out a surprising co-existence of both real discrete spectrum and a single SS in a fixed potential. Nevertheless, the complex Scarf II is not known to be pseudo-Hermitian ($η^{-1} Hη=H^\dagger$) under a metric of the type $η(x)$, so far.

quant-ph↗
arXiv2020

An interesting track for the Brachistochrone

If a particle has to fall first vertically 1 m from A and then move horizontally 1 m to B, it takes a time $t(=τ_1+τ_2=τ_3=3/\sqrt{2g})=0.67$ s. Under gravity and without friction, if it sides down on a linear track inclined at $45^0$ between two points A and B of 1 m height, it takes time $t(=τ_4=2/\sqrt{g})=0.63$ s. Between these two extremes, historically, Bernoulli (1718) proved that the fastest track between these points A and B is cycloid with the least time of descent $t=τ_B=0.58$ s. Apart from other interesting cases, here we study the frictionless motion of a particle/bead on an interesting track/wire between A and B given by $y(x)=(1-x^ν)^{1/ν}.$ For $ν> 1$ the track becomes convex and $t>>τ_4$, and when $ν>1.22$, the motion with zero initial speed is not possible. We find that when $ν\in (0.09653, 0.31749), τ_4<t <τ_3$ and when $ν\in ( 0.31749, 1),τ_B < t < τ_4$. But most remarkably, the concave curve becomes very steep/deep if $ν\in (0, ν_c=0.09653)$, then $t=0.2258$ s $< τ_B$, this is as though a particle would travel 1 meter horizontally with a speed equal $\sqrt{2g}$ m/sec to take the time ($=1/\sqrt{2g}=τ_2) < τ_B$. The function $t(ν$) suffers a jump discontinuity at $ν=ν_c$, we offer some resolution.

physics.class-ph↗
arXiv2020

Families of curves orthogonal to the lines $y=mx-2m-m^3$

The family of lines $y=mx-2m-m^3$, are well known to be normal to the parabola $y^2=4x$. However, this family of lines is normal to a family of curves of which this parabola is just one member. Here, by solving an interesting first order and third degree ODE, we bring out these curves. The resulting one set of curves are "parabola-like" but non-standard ones and the other family is not even "parabola like".

math.GM↗
arXiv2020

PT-symmetric potentials with imaginary asymptotic saturation

We point out that PT-symmetric potentials $V_{PT}(x)$ having imaginary asymptotic saturation: $V_{PT}(x=\pm \infty) =\pm i V_1, V_1 \in \Re$ are devoid of scattering states and spectral singularity. We show the existence of real (positive and negative) discrete spectrum both with and without complex conjugate pair(s) of eigenvalues (CCPEs). If the states are arranged in the ascending order or real part of discrete eigenvalues, the initial states have few nodes but latter ones oscillate fast. Both real and imaginary parts of $ψ(x)$ vanish asymptotically, $|ψ(x)|$ for the CCPEs are asymmetric and for real energies these are symmetric about origin. For CCPEs $E_{\pm}$ the eigenstates $ψ_{\pm}$ follow an interesting property that $|ψ_+(x)|= N |ψ_-(-x)|, N \in \Re^+$. We remark that, the fast oscillating real discrete energy states discussed are likely to be confused with: reflectionless states, one dimensional version of von Neumann states of Hermitian and spectral singularity state of complex PT-symmetric potentials.

quant-ph↗
arXiv2020

Low reflection at zero or low-energies in the well-barrier scattering potentials

Probability of reflection $R(E)$ off a finite attractive scattering potential at zero or low energies is ordinarily supposed to be 1. However, a fully attractive potential presents a paradoxical result that $R(0)=0$ or $R(0)<1$, when an effective parameter $q$ of the potential admits special discrete values. Here, we report another class of finite potentials which are well-barrier (attractive-repulsive) type and which can be made to possess much less reflection at zero and low energies for a band of low values of $q$. These well-barrier potentials have only two real turning points for $E \in(V_{min}, V_{max})$, excepting $E=0$. We present two exactly solvable and two numerically solved models to confirm this phenomenon.

quant-ph↗
arXiv2019

Symmetric Fermi-type potential

We utilize the amenability of the Fermi-type potential profile in Schr{ö}dinger equation to construct a symmetric one dimensional well as $V(x){=}{-}U_n/[1+\exp[(|x|{-}a)/b]], ~ U_n{=}V_n[1+\exp[-a/b]]$. We define $α=a/b, ~β_n {=}b\sqrt{2m U_n}/\hbar$, we find $β_n$ values for which critically the well has $n$-node half bound state at $E{=}0$. Consequently, this fixed well has $n$ number of bound states. Also we obtain a semi-classical expression ${\cal G}(α,β)$ such that the Fermi well has either $[\cal G]$ or $[{\cal G}]+1$ number of bound states. Here $[.]$ indicates the integer part. We also confirm the consistency of $\cal G$ with the number of s-wave neutron energy levels in a central ($x\in (0,\infty))$ Fermi potential well.

quant-ph↗
arXiv2019

Solvable model of bound states in the continuum (BIC) in one dimension

Historically, most of the quantum mechanical results have originated in one dimensional model potentials. However, Von-Neumann's Bound states in the Continuum (BIC) originated in specially constructed, three dimensional, oscillatory, central potentials. One dimensional version of BIC has long been attempted, where only quasi-exactly-solvable models have succeeded but not without instigating degeneracy in one dimension. Here, we present an exactly solvable bottomless exponential potential barrier $V(x)=-V_0[\exp(2|x|/a)-1]$ which for $E V_0$, there is again a continuum of complex scattering solutions $ψ(x)$ whose real and imaginary parts though solutions of Schr{ö}dinger equation yet their parities cannot be ascertained as $Cψ(x)$ is also a solution where $C$ is an arbitrary complex non-real number.

quant-ph↗
arXiv2019

Expectation value of $p^6$ in continuous two-piece symmetric potential wells

Earlier, potentials like square well and several other half-potential wells with discontinuous jump have been found to have the expectation value $<\! p^6 \!>$ to be divergent for all bound states. Here, we consider two-piece symmetric potential wells to prove and demonstrate that in them the expectation value of $p^6$ diverges for even states and converges for odd states. Here, $p$ denotes momentum. We also present three exactly solvable models.

quant-ph↗
arXiv2018

Three types of discrete energy eigenvalues in complex PT-symmetric scattering potentials

For complex PT-symmetric scattering potentials (CPTSSPs) $V(x)= V_1 f_{even}(x) + iV_2 f_{odd}(x), f_{even}(\pm \infty) = 0 = f_{odd}(\pm \infty), V_1,V_2 \in \Re $, we show that complex $k$-poles of transmission amplitude $t(k)$ or zeros of $1/t(k)$ of the type $\pm k_1+ik_2, k_2\ge 0$ are physical which yield three types of discrete energy eigenvalues of the potential. These discrete energies are real negative, complex conjugate pair(s) of eigenvalues (CCPEs: ${\cal E}_n \pm i γ_n$) and real positive energy called spectral singularity (SS) at $E=E_*$ where the transmission and reflection co-efficient of $V(x)$ become infinite for a special critical value of $V_2=V_*$. Based on four analytically solvable and other numerically solved models, we conjecture that a parametrically fixed CPTSSP has at most one SS. When $V_1$ is fixed and $V_2$ is varied there may exist Kato's exceptional point(s) $(V_{EP})$ and critical values $V_{*m}, m=0,1,2,..$, so when $V_2$ crosses one of these special values a new CCPE is created. When $V_2$ equals a critical value $V_{*m}$ there exist one SS at $E=E_*$ along with $m$ or more number of CCPEs. Hence, this single positive energy $E_*$ is the upper (or rough upper) bound to the CCPEs: ${\cal E}_l \lessapprox E_*$, here ${\cal E}_l$ corresponds to the last of CCPEs. If $V(x)$ has Kato's exceptional points (EPs: $V_{EP1}<V_{EP2}<V_{EP3}<...<V_{EPl}$), the smallest of critical values $V_{*m}$ is always larger than $V_{EPl}$. Hence, in a CPTSSP, real discrete eigenvalue(s) and the SS are mutually exclusive whereas CCPEs and the SS can co-exist .

quant-ph↗
arXiv2018

Pearson's correlation coefficient in the theory of networks: A comment

In statistics, the Pearson correlation coefficient $r_{x,y}$ determines the degree of linear correlation between two variables and it is known that $-1 \le r_{x,y} \le 1$. In the theory of networks, a curious expression proposed in [PRL {\bf 89} 208701 (2002)] for degree-degree correlation coefficient $r_{j_i,k_i}, i\in [1,M]$ has been in use. We realize that the suggested form is the conventional Pearson's coefficient for $\{(j_i,k_i), (k_i,j_i)\}$ for $2M$ data points and hence it is rightly dedicated to undirected networks.

cond-mat.dis-nn↗
arXiv2018

Divergence of $\langle p^6\rangle$ in discontinuous potential wells

The surprising divergence of the expectation value $<\!p^6\!>$ for the square well potential is known. Here, we prove and demonstrate the divergence of $<\!p^6\!>$ in potential wells which have a finite jump discontinuity; apart from the square-well two-piece half-potentials wells are examples. These half-potential wells can be expressed as $V(x)=-U(x) Θ(x)$, where $Θ(x)$ is the Heaviside step function. $U(x)$ are continuous and differentiable functions with minimum at $x=0$ and which may or not vanish as $x\sim \infty$.

quant-ph↗
arXiv2018

Coherent scattering from semi-infinite non-Hermitian potentials

When two identical (coherent) beams are injected at a semi-infinite non-Hermitian medium from left and right, we show that both reflection $(r_L,r_R)$ and transmission $(t_L,t_R)$ amplitudes are non-reciprocal. In a parametric domain, there exists Spectral Singularity (SS) at a real energy $E=E_*$ and the determinant of the time-reversed two port S-matrix i.e., $|\det(S)|=|t_L t_R-r_L r_R|$ vanishes sharply at $E=E_*$ displaying the phenomenon of Coherent Perfect Absorption (CPA). In the complimentary parametric domain, the potential becomes either left or right reflectionless at $E=E_z$. But we rule out the existence of Invisibility despite $r_R(E_i)=0$ and $t_R(E_i)=1$ in these new models. We present two simple exactly solvable models where the expressions for $E_*$, $E_z$, $E_i$ and the parametric conditions on the potential have been obtained in explicit and simple forms. Earlier, the novel phenomena of SS and CPA have been found to occur only in the scattering complex potentials which are spatially localized (vanish asymptotically) and having $t_L=t_R$.

quant-ph↗
arXiv2018

Expectation values of $p^2$ and $p^4$ in the square well potential

Position and momentum representations of a wavefunction $ψ(x)$ and $ϕ(p)$, respectively are physically equivalent yet mathematically in a given case one may be easier or more transparent than the other. This disparity may be so much so that one has to device a special strategy to get the quantity of interest in one of them. We revisit finite square well (FSW) in this regard. Circumventing the the problems of discontinuity of second and higher derivatives of $ψ(x)$ we obtain simple analytic expressions of $<\!p^2\!>$ and $<\!p^4\!>$. But it is the surprising fall-off of $ϕ(p)$ as $p^{-6}$ that reveals and restricts $<\!p^s\!>$ to be finite and non-zero only for $s=2,4$. In finding $<\!p^s\!>(s=2,4)$ from $ϕ(p)$, $p$-integrals are improper which for time-being, have been evaluated numerically to show the agreement between two representations.

quant-ph↗
arXiv2017

Pseudo-symmetric random matrices: semi-Poisson and sub-Wigner statistics

Real non-symmetric matrices may have either real or complex conjugate eigenvalues. These matrices can be seen to be pseudo-symmetric as $ηM η^{-1} = M^t$, where the metric $η$ could be secular (a constant matrix) or depending upon the matrix elements of $M$. Here, we construct ensembles of a large number $N$ of pseudo-symmetric $n \times n$ ($n$ large) matrices using ${\cal N}$ $(n(n+1)/2 \le {\cal N} \le n^2)$ independent and identically distributed (iid) random numbers as their elements. Based on our numerical calculations, we conjecture that for these ensembles the Nearest Level Spacing Distributions (NLSDs: $p(s)$) are sub-Wigner as $p_{abc}(s)=a s e^{-bs^c} (0<c <2)$ and the distributions of their eigenvalues fit well to $D(ε)= A[\mbox{tanh}\{(ε+B)/C \}-\mbox{tanh}\{(ε-B)/C\}]$ (exceptions also discussed). These sub-Wigner NLSD are encountered in Anderson metal-insulator transition and topological transitions in a Josephson junction. Interestingly, $p(s)$ for $c=1$ is called semi-Poisson and we show that it lies close to the form $p(s)=0.59 s K_0(0.45 s^2)$ derived for the case of $2 \times 2$ pseudo-symmetric matrix where the eigenvalues are most aptly conditionally real: $E_{1,2}=a \pm \sqrt{b^2-c^2}$ which represent characteristic coalescing of eigenvalues in PT(Parity-Time)-symmetric systems.

quant-ph↗