158 research outputs found
Sextic anharmonic oscillators and orthogonal polynomials
Under certain constraints on the parameters a, b and c, it is known that
Schroedinger's equation
-y"(x)+(ax^6+bx^4+cx^2)y(x) = E y(x), a > 0, with the sextic anharmonic
oscillator potential is exactly solvable. In this article we show that the
exact wave function y is the generating function for a set of orthogonal
polynomials P_n^{(t)}(x) in the energy variable E. Some of the properties of
these polynomials are discussed in detail and our analysis reveals scaling and
factorization properties that are central to quasi-exact solvability. We also
prove that this set of orthogonal polynomials can be reduced,by means of a
simple scaling transformation, to a remarkable class of orthogonal polynomials,
P_n(E)=P_n^{(0)}(E) recently discovered by Bender and Dunne.Comment: 11 page
Towards Sustainable and Intelligent Machining:Energy Footprint and Tool Condition Monitoring for Media-Assisted Processes
Reducing energy consumption is a necessity towards achieving the goal of net-zero manufacturing. In this paper, the overall energy footprint of machining Ti-6Al-4V using various cooling/lubrication methods is investigated taking the embodied energy of cutting tools and cutting fluids into account. Previous studies concentrated on reducing the energy consumption associated with the machine tool and cutting fluids. However, the investigations in this study show the significance of the embodied energy of cutting tool. New cooling/lubrication methods such as WS2-oil suspension can reduce the energy footprint of machining through extending tool life. Cutting tools are commonly replaced early before reaching their end of useful life to prevent damage to the workpiece, effectively wasting a portion of the embodied energy in cutting tools. A deep learning method is trained and validated to identify when a tool change is required based on sensor signals from a wireless sensory toolholder. The results indicated that the network is capable of classifying over 90% of the tools correctly. This enables capitalising on the entirety of a tool’s useful life before replacing the tool and thus reducing the overall energy footprint of machining processes
Towards Sustainable and Intelligent Machining:Energy Footprint and Tool Condition Monitoring for Media-Assisted Processes
Reducing energy consumption is a necessity towards achieving the goal of net-zero manufacturing. In this paper, the overall energy footprint of machining Ti-6Al-4V using various cooling/lubrication methods is investigated taking the embodied energy of cutting tools and cutting fluids into account. Previous studies concentrated on reducing the energy consumption associated with the machine tool and cutting fluids. However, the investigations in this study show the significance of the embodied energy of cutting tool. New cooling/lubrication methods such as WS2-oil suspension can reduce the energy footprint of machining through extending tool life. Cutting tools are commonly replaced early before reaching their end of useful life to prevent damage to the workpiece, effectively wasting a portion of the embodied energy in cutting tools. A deep learning method is trained and validated to identify when a tool change is required based on sensor signals from a wireless sensory toolholder. The results indicated that the network is capable of classifying over 90% of the tools correctly. This enables capitalising on the entirety of a tool’s useful life before replacing the tool and thus reducing the overall energy footprint of machining processes
Study of the Generalized Quantum Isotonic Nonlinear Oscillator Potential
We study the generalized quantum isotonic oscillator Hamiltonian given by H=−d2/dr2+l(l+1)/r2+w2r2+2g(r2−a2)/(r2+a2)2, g>0. Two approaches are explored. A method for finding
the quasipolynomial solutions is presented, and explicit expressions for these polynomials are given,
along with the conditions on the potential parameters. By using the asymptotic iteration method,
we show how the eigenvalues of this Hamiltonian for arbitrary values of the parameters g, w, and a
may be found to high accuracy
Asymptotic iteration method for eigenvalue problems
An asymptotic interation method for solving second-order homogeneous linear
differential equations of the form y'' = lambda(x) y' + s(x) y is introduced,
where lambda(x) \neq 0 and s(x) are C-infinity functions. Applications to
Schroedinger type problems, including some with highly singular potentials, are
presented.Comment: 14 page
Coulomb plus power-law potentials in quantum mechanics
We study the discrete spectrum of the Hamiltonian H = -Delta + V(r) for the
Coulomb plus power-law potential V(r)=-1/r+ beta sgn(q)r^q, where beta > 0, q >
-2 and q \ne 0. We show by envelope theory that the discrete eigenvalues
E_{n\ell} of H may be approximated by the semiclassical expression
E_{n\ell}(q) \approx min_{r>0}\{1/r^2-1/(mu r)+ sgn(q) beta(nu r)^q}.
Values of mu and nu are prescribed which yield upper and lower bounds.
Accurate upper bounds are also obtained by use of a trial function of the form,
psi(r)= r^{\ell+1}e^{-(xr)^{q}}. We give detailed results for
V(r) = -1/r + beta r^q, q = 0.5, 1, 2 for n=1, \ell=0,1,2, along with
comparison eigenvalues found by direct numerical methods.Comment: 11 pages, 3 figure
Energies and wave functions for a soft-core Coulomb potential
For the family of model soft Coulomb potentials represented by V(r) =
-\frac{Z}{(r^q+\beta^q)^{\frac{1}{q}}}, with the parameters
Z>0, \beta>0, q \ge 1, it is shown analytically that the potentials and
eigenvalues, E_{\nu\ell}, are monotonic in each parameter. The potential
envelope method is applied to obtain approximate analytic estimates in terms of
the known exact spectra for pure power potentials. For the case q =1, the
Asymptotic Iteration Method is used to find exact analytic results for the
eigenvalues E_{\nu\ell} and corresponding wave functions, expressed in terms of
Z and \beta. A proof is presented establishing the general concavity of the
scaled electron density near the nucleus resulting from the truncated
potentials for all q. Based on an analysis of extensive numerical calculations,
it is conjectured that the crossing between the pair of states
[(\nu,\ell),(\nu',\ell')], is given by the condition \nu'\geq (\nu+1) and \ell'
\geq (\ell+3). The significance of these results for the interaction of an
intense laser field with an atom is pointed out. Differences in the observed
level-crossing effects between the soft potentials and the hydrogen atom
confined inside an impenetrable sphere are discussed.Comment: 13 pages, 5 figures, title change, minor revision
Solutions for certain classes of Riccati differential equation
We derive some analytic closed-form solutions for a class of Riccati equation
y'(x)-\lambda_0(x)y(x)\pm y^2(x)=\pm s_0(x), where \lambda_0(x), s_0(x) are
C^{\infty}-functions. We show that if \delta_n=\lambda_n
s_{n-1}-\lambda_{n-1}s_n=0, where \lambda_{n}=
\lambda_{n-1}^\prime+s_{n-1}+\lambda_0\lambda_{n-1} and
s_{n}=s_{n-1}^\prime+s_0\lambda_{k-1}, n=1,2,..., then The Riccati equation has
a solution given by y(x)=\mp s_{n-1}(x)/\lambda_{n-1}(x). Extension to the
generalized Riccati equation y'(x)+P(x)y(x)+Q(x)y^2(x)=R(x) is also
investigated.Comment: 10 page
Criterion for polynomial solutions to a class of linear differential equation of second order
We consider the differential equations y''=\lambda_0(x)y'+s_0(x)y, where
\lambda_0(x), s_0(x) are C^{\infty}-functions. We prove (i) if the differential
equation, has a polynomial solution of degree n >0, then \delta_n=\lambda_n
s_{n-1}-\lambda_{n-1}s_n=0, where \lambda_{n}=
\lambda_{n-1}^\prime+s_{n-1}+\lambda_0\lambda_{n-1}\hbox{and}\quad
s_{n}=s_{n-1}^\prime+s_0\lambda_{k-1},\quad n=1,2,.... Conversely (ii) if
\lambda_n\lambda_{n-1}\ne 0 and \delta_n=0, then the differential equation has
a polynomial solution of degree at most n. We show that the classical
differential equations of Laguerre, Hermite, Legendre, Jacobi, Chebyshev (first
and second kind), Gegenbauer, and the Hypergeometric type, etc, obey this
criterion. Further, we find the polynomial solutions for the generalized
Hermite, Laguerre, Legendre and Chebyshev differential equations.Comment: 12 page
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