## WKB Approximation of the Nonlinear Schr¨odinger-Newton EquationsBook 2.29 MB | Ebook Pages: 61WKB Approximation of the Nonlinear Schr¨odinger-Newton Equations Carsten Hartmann† and Heinz-Ju¨rgen Schmidt‡ †Free University Berlin, Institute of Mathematics II |

## The Nonoscillatory CaseBook 5.63 MB | Ebook Pages: 78WKB (Wentzel-Kramers-Brillouin) Approximation 1. We apply the WKB method to approximate solutions to equations of the form ε2y00 + q(x)y = 0, ε ˝ 1, (1) |

## The WKB Approximation Lectures Nine and Ten The WKB ApproximationBook 6.77 MB | Ebook Pages: 220The WKB Approximation Lectures Nine and Ten The WKB Approximation The WKB method is a powerful tool to obtain solutions for many physIcal problems. |

## The WKB Approximation Lecture Eleven Turning PointBook 5.82 MB | Ebook Pages: 97The WKB Approximation Lecture Eleven Turning Point As we have mentioned, the WKB approximation is useful in problems of wave propagation. In |

## Complex WKB Solutions and Hamiltonian Systems on Riemann SurfacesBook 6.77 MB | Ebook Pages: 83(see Ablowitz and Segur [1981]) for small>. and show that it coincides with the WKB approximation for the usual SchrOdinger equation (2.1). Then we will establish a link |

## Ch09 Approximate solutionsBook 4.86 MB | Ebook Pages: 168and Brillouin (1926) and is called the WKB method or WKB approximation. This method provides approximate wave functions in one-dimensional problems. |

## Complex WKB Approximations in Graphene Electron-Hole Waveguides inBook 5.34 MB | Ebook Pages: 197method of Gaussian beams summation is efﬁcient for construction of WKB approximation descriBing electron-hole motion in magnetic ﬁeld and any scalar potential |

## Chapter 11 WKB and the path integralBook 6.77 MB | Ebook Pages: 108The original application of the WKB approximation is the derivation of the Bohr–Sommerfeld quantization condition. If we begin our considerations, for simplicity, with |

## 1 Resonant tunneling structuresBook 3.24 MB | Ebook Pages: 101(0) e ( ) (0) e h The tunneling probability in the WKB approximation is given by: ∫ [] = = B ∗ 0 – L 2 –1 2 ( )– d x m U x E x T e h |

## Tunneling in III-V NitridesBook 3.62 MB | Ebook Pages: 192coefficient is calculated for interband tunnel diode using WKB approximation by potential barrier method and E-k diagram. In order to get substantial amount of tunneling |

## Wave Reflection over Slowly Varying Bathymetry modeled byBook 4.48 MB | Ebook Pages: 160The comparison between the analytIcal solution of Reflection WKB approximation with the numerical solution is shown in fig. (4)-(5). The total simulation time is 25 minutes. |

## Spectral properties of solutions of hypergeometric-typeBook 5.91 MB | Ebook Pages: 249This study will be done within the so-called semiclassIcal (or WKB) approximation [7, 10, 14]. As it is described in Section 2, a recent result [2] based on this method |

## G.P.Gilfoyle Physics Department University of Richmond, VirginiaBook 2.1 MB | Ebook Pages: 67and ¯h is Planck’s constant divided by 2π. Potential barriers of arbitrary shape are then treated using the Wentzel, Kramers, Brillouin (WKB) approximation.1−7 This |

## Instructor: Phone: Email: TABook 5.25 MB | Ebook Pages: 173• apply several techniques such as perturbation theory, WKB approximation and adiabatic approach to study quantum particles under external fields |

## The mode-coupling Liouville Green approximation for a twoBook 3.34 MB | Ebook Pages: 137The Liouville–Green @or Wentzel–Kramers–Brillouin ~WKB!# approximation for the two-dimensional cochlear mechanics problem disagrees with the ﬁnite-difference |

## Free electron lasers with slowly-varying beam and undulator parametersBook 5.53 MB | Ebook Pages: 74and undulator parameters is developed using the WKB approximation. The the-ory is applied to study the performance of a self-AMPLiﬂed spontaneous emission |

## Direct tunneling hole currents through ultrathin gate oxides inBook 3.43 MB | Ebook Pages: 237probability.6–8,12,18 In the WKB approximation, it is crucial to use an appropriate hole dispersion in the SiO2 band gap. All previous Models assume a parabolic dispersion |

## Effect of radiation losses on hotspot formation and propagation inBook 5.05 MB | Ebook Pages: 247expected from the nature of the WKB approximation. On the other hand, it can be seen from Fig. 3(a) tha the agreement is uniform over a wide range of values |

## WORMHOLES WITH A PAST*Book 4.39 MB | Ebook Pages: 175For a euclidean solution to represent a WKB approximation of a tunneling between two classIcally allowed regions, the euclidean |

## No-sticking effect and quantum reﬂection in ultracold collisionsBook 4.86 MB | Ebook Pages: 161feet of the WKB approximation, which breaks down in the long-range attractive part of the potential at low energy. Very far out, the WKB is good even for low energy |

## Q u an tu m M ech an icsBook 5.05 MB | Ebook Pages: 177We can make life a bit easier by using the quantization condition derived from the WKB approximation. Since we require the wavefunction to vanish exactly at x= 0, we have: |

## PHY 407 QUANTUM MECHANICS Fall 05 Problem set 1 Due Sep 15 2005Book 2.67 MB | Ebook Pages: 182Use the WKB approximation to ﬁnd a formula for the number of bound states of energy less than E. In particular, estimate the total number bound states ( i.e., the number |

## Asymptotic iteration method applied to bound-state problems withBook 1.72 MB | Ebook Pages: 243One of the methods mentioned above, the WKB approximation, is for solving the SchrOdinger equation by expanding the wave function in the power series of¨ ~. |

## Electron tunneling through the repulsive Coulomb barrier inBook 1.72 MB | Ebook Pages: 100The observed tunneling effect and spectral shift are explained using WKB 22n2 approximation based on a Model potential with a square well at short range and a repulsive |

## Quantum Mechanics - II PHYS 4308Book 2.29 MB | Ebook Pages: 62common and powerful tools in quantum mechanics and will be studied next. The WKB approximation will be followed by time-dependent perturbation theory where we will cover |

## Analytical Models for Quantized Sub-Band Energy Levels andBook 1.43 MB | Ebook Pages: 122(17) to determine the WKB integral I values in (14) gives a better result compare to the triangular approximation at strong inversion region. Figure 2 shows the numerIcal |

## Quantum Picture of the Josephson JunctionBook 7.15 MB | Ebook Pages: 157The results match the results from WKB approximation with 3 TT21/~10 and 2 TT32/~10, but 2 TT10/~10in this calculation is slightly different from |

## Nanofocusing of Optical Energy in Tapered PlasmonicWaveguidesBook 1.72 MB | Ebook Pages: 233equations in eikonal (WKB) approximation in the form: E z r;z R r I 0 k 0 mr r R BK 0 k 0 dr ;E z r; z R r i n m I 1 k 0 mr r R i n d BK 1 k 0 dr ; (4) where B I |

## Modeled Tunnel Currents For High Dielectric Constant DielectricsBook 4.48 MB | Ebook Pages: 234using the WKB approximation as [2], [13] (6) where is the insulator effective mass which is assumed constant. The integration in (6) is over all real values of |

## Year 1 Progress Report for NASA AISR Grant NNG04GP83G: A NumericalBook 4.67 MB | Ebook Pages: 197The FDTD scattering component and the WKB-approximation propagation component were combined to calculate the reflected signals received by orbital GPR. |

## Physics 216. Quantum Mechanics. Professor Dine Spring, 2004. SyllabusBook 5.34 MB | Ebook Pages: 103The WKB approximation, tunneling, perhaps a little bit of path integration. (Shankar 16, Sakurai 5.4, also pp. 104-109 from chapter 2. Supplementary handouts). |

## 9 Reﬂection, Absorption, and Mode ConversionBook 4.29 MB | Ebook Pages: 127When k2(x) is slowly varying, i.e., |k′′| ≪ |kk′| and |k′| ≪ |k2|, the WKB approximation can be used. The solution is given by E = (const. |

## Coherent and Incoherent Scattering by Oceanic BubblesBook 3.91 MB | Ebook Pages: 150rays, a straightforward application of the lowest-order WKB approximation to the Helmholtz equation (1) for the coherent field (p) generated by a unit point source |

## Tunneling in long-wavelength infrared HgCdTe photodiodesBook 6.39 MB | Ebook Pages: 158lated from the WKB approximation.The barrier height is the bandgap for BTB, 1 whereasinTAT calculations it is determined by the energies of trap states. |

## Modeling CMOS tunneling currents through ultrathin gate oxide dueBook 1.62 MB | Ebook Pages: 242the WKB approximation [11], Schuegraf et al. have derived a simple analytIcal formula to represent direct-tunneling through a trapezoidal barrier [1], [2]. |

## IEEE Electron Device LettersBook 6.39 MB | Ebook Pages: 216can be estimated using the WKB approximation [15], [16]. - (1) where and are the respective widths of sub-oxide regions (in Å) with and without Interface nitridation, and |

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