Quasilinearization Method and Summation of the WKB Series

Solutions obtained by the quasilinearization method (QLM) are compared with the WKB solutions. Expansion of the $p$-th QLM iterate in powers of $\hbar$ reproduces the structure of the WKB series generating an infinite number of the WKB terms with the first $2^p$ terms reproduced exactly. The QLM quantization condition leads to exact energies for the P\"{o}schl-Teller, Hulthen, Hylleraas, Morse, Eckart potentials etc. For other, more complicated potentials the first QLM iterate, given by the closed analytic expression, is extremely accurate. The iterates converge very fast. The sixth iterate of the energy for the anharmonic oscillator and for the two-body Coulomb Dirac equation has an accuracy of 20 significant figures

On Exactness Of The Supersymmetric WKB Approximation Scheme

Exactness of the lowest order supersymmetric WKB (SWKB) quantization condition $\int^{x_2}_{x_1} \sqrt{E-\omega^2(x)} dx = n \hbar \pi$, for certain potentials, is examined, using complex integration technique. Comparison of the above scheme with a similar, but {\it exact} quantization condition, $\oint_c p(x,E) dx = 2\pi n \hbar$, originating from the quantum Hamilton-Jacobi formalism reveals that, the locations and the residues of the poles that contribute to these integrals match identically, for both of these cases. As these poles completely determine the eigenvalues in these two cases, the exactness of the SWKB for these potentials is accounted for. Three non-exact cases are also analysed; the origin of this non-exactness is shown to be due the presence of additional singularities in $\sqrt{E-\omega^2(x)}$, like branch cuts in the $x-$plane.Comment: 11 pages, latex, 1 figure available on reques

Periodic Quasi - Exactly Solvable Models

Various quasi-exact solvability conditions, involving the parameters of the periodic associated Lam{\'e} potential, are shown to emerge naturally in the quantum Hamilton-Jacobi approach. It is found that, the intrinsic nonlinearity of the Riccati type quantum Hamilton-Jacobi equation is primarily responsible for the surprisingly large number of allowed solvability conditions in the associated Lam{\'e} case. We also study the singularity structure of the quantum momentum function, which yields the band edge eigenvalues and eigenfunctions.Comment: 11 pages, 5 table

Calculation of Band Edge Eigenfunctions and Eigenvalues of Periodic Potentials through the Quantum Hamilton - Jacobi Formalism

We obtain the band edge eigenfunctions and the eigenvalues of solvable periodic potentials using the quantum Hamilton - Jacobi formalism. The potentials studied here are the Lam{\'e} and the associated Lam{\'e} which belong to the class of elliptic potentials. The formalism requires an assumption about the singularity structure of the quantum momentum function $p$, which satisfies the Riccati type quantum Hamilton - Jacobi equation, $p^{2} -i \hbar \frac{d}{dx}p = 2m(E- V(x))$ in the complex $x$ plane. Essential use is made of suitable conformal transformations, which leads to the eigenvalues and the eigenfunctions corresponding to the band edges in a simple and straightforward manner. Our study reveals interesting features about the singularity structure of $p$, responsible in yielding the band edge eigenfunctions and eigenvalues.Comment: 21 pages, 5 table

Quantum Hamilton-Jacobi analysis of PT symmetric Hamiltonians

We apply the quantum Hamilton-Jacobi formalism, naturally defined in the complex domain, to a number of complex Hamiltonians, characterized by discrete parity and time reversal (PT) symmetries and obtain their eigenvalues and eigenfunctions. Examples of both quasi-exactly and exactly solvable potentials are analyzed and the subtle differences, in the singularity structures of their quantum momentum functions, are pointed out. The role of the PT symmetry in the complex domain is also illustrated.Comment: 11 page

Accuracy of Semiclassical Methods for Shape Invariant Potentials

We study the accuracy of several alternative semiclassical methods by computing analytically the energy levels for many large classes of exactly solvable shape invariant potentials. For these potentials, the ground state energies computed via the WKB method typically deviate from the exact results by about 10%, a recently suggested modification using nonintegral Maslov indices is substantially better, and the supersymmetric WKB quantization method gives exact answers for all energy levels.Comment: 7 pages, Latex, and two tables in postscrip

The Estimation of Helicopter Pilot Workload Using Inverse Simulation: Longitudinal Manoeuvre Analysis. Internal Report No. 9625

In the preceding report the concept of estimating pilot workload using inverse simulation was introduced. The report examined the ADS-33C defined Rapid Side-step Mission Task Element (MTE), and illustrated how various quickness parameters could be obtained from the lateral cyclic pitch and stick displacement time histories. These quickness parameters were plotted on charts and it was shown how the resulting plots could be used to discriminate between two dissimilar helicopter configurations, or identify which manoeuvres were more aggressive and would probably lead to a higher level of workload being placed upon the pilot. The intention of this report is to provide a supplementary study to the previous one by analysing another linear repositioning manoeuvre, the Rapid Acceleration / Deceleration or Quick-hop MTE. The longitudinal cyclic channel will be investigated in terms of pitch and stick displacement and the equivalent quickness parameters calculated and plotted on charts. A final study mirroring the previous one, on control system influence by the introduction of a Stability and Control Augmentation System, (SCAS) and the alteration of the longitudinal cyclic actuator constant will also be carried out

The Estimation of Precision Pilot Model Parameters Using Inverse Simulation. Internal Report No. 9706

The practice of using mathematical models to simulate pilot behaviour in one-axis stabilisation tasks is a well known conventional simulation problem. In this report a system is developed whereby a mathematical model of a pilot is used as the controller of a rudimentary helicopter model. The main differences between this and other similar scenarios that have been found in the literature are that firstly, inverse simulation is used to provide results that are used as the forcing functions in the model of the pilot/helicopter system, and secondly a constrained optimisation routine is utilised to obtain values for the parameters within the pilot model itself. It will be shown that as the pilot is required to fly different manoeuvres, defined by standards set by the United States Army, or indeed if the severity of the set manoeuvres is varied, the pilot is required to adjust certain human parameters to fly the manoeuvre in a superlative manner. The report considers initially the pilot and helicopter models and subsequently analyses the system as a whole, illustrating how the pilot model can change depending on the circumstances

The Estimation of Helicopter Pilot Workload Using Inverse Simulation. Internal Report No. 9624

In the first instance this report describes the means by which inverse simulation can be used as a pilot workload estimation tool. An alternative approach to defining the mathematical model of the ADS-33 Rapid Side-step Mission Task Element (MTE) is presented and is used to drive various inverse simulation runs. Studies are conducted into three varying aggression side-step MTEs and the comparison of two dissimilar helicopter configurations based on the Westland Lynx, simulated using the same side-step. It is shown how the resulting time-histories and quickness charts can be utilised in pilot workload and handling qualities estimation. A third quickness parameter associated with the lateral cyclic stick displacements required to fly the side-step MTEs is introduced and is shown to be capable of discriminating between the pilot workload required for each side-step and vehicle configuration. The latter study in the report presents the preliminary findings on the effects of workload by firstly, introducing a Stability and Control Augmentation System and secondly investigating the effects of altering the value of the lateral cyclic actuator time constant

On the Development of Multiple Manoeuvre Mission Sequences for Inverse Simulation. Internal Report No. 9802

As part of the continuing programme of work and collaboration between the Defence Evaluation and Research Agency (DERA) and Glasgow University (GU), the author was invited to attend the final phase of flight simulation trials entitled ‘TWINS’ at DERA, Bedford; using the Advanced Flight Simulator (AFS) large motion system. The precise nature and details of the five-day trial are given in [1] but the main thrust of the trial was essentially divided into two areas: 1. The simulation of American Design Standard (ADS) Mission Task Elements (MTEs) using a software image database of Coltishall airfield with the appropriate ADS-33 visual cues. 2. The simulation of a mission sequence based on the Haxton Down software image database which comprised fourteen individual tasks. The tasks were either based on ADS MTEs or Nap-of-the-earth (NOE) flight. A full description of the manoeuvre elements is given in Appendix A of [1]. The inverse simulation package HELINV at GU contains a library of manoeuvres based both on ADS MTEs and NOE flight. However, the manoeuvres are separate and individual and until recently it was not possible to run a simulation of combinations of two or more manoeuvres. A request was put forward to develop a method whereby it was possible to choose several elements (MTE or NOE) from the manoeuvre menu and piece them together to form what has been termed a ‘mini-mission sequence’ and then inverse simulate the mission as a whole. This report describes that development and presents the results from several simulated mission runs