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Field Emission and Vacuum Devices
Published in John P. Xanthakis, Electronic Conduction, 2020
Both equations 6.17 and 6.19 present a mathematical problem at points x such that V=E, i.e. where the potential energy is equal to the total energy. Then the kinetic energy is zero and hence the classical velocity is zero. From both equations 6.17 and 6.19 we deduce that the wavefunction Ψ goes to infinity. In this case the WKB approximation breaks down. The points in space at which this happens are shown in figure 6.1a and are designated “turning points” because classically the velocity would have changed direction at these points. These points are crucial because it is at these points we have to calculate the wavefunction to obtain the transmission of a wave through a barrier such as the one shown in figure 6.1a. The transmission coefficient defined by T=ΨtransmittedΨincident2×vtransmittedvincident
Fundamental Light–Tissue Interactions: Light Scattering and Absorption
Published in Vadim Backman, Adam Wax, Hao F. Zhang, A Laboratory Manual in Biophotonics, 2018
Vadim Backman, Adam Wax, Hao F. Zhang
If a particle is much larger than the wavelength and the phase shift is not negligible, the first-order Born approximation is no longer valid. In tissue optics, this is pertinent to a case when we consider scattering by large scatterers such as individual cells or large organelles such as nuclei. In these cases, the scattering amplitude and the scattering cross-section can be estimated using the Wentzel–Kramers–Brillouin approximation (WKB). Just like the Born approximation, the WKB approximation has been developed and widely applied to solve quantum mechanical scattering problems. In optics, several authors have independently arrived at essentially the same approximation; thus, the method has acquired many different names, including the Van de Hulst, the anomalous diffraction, or the interior wave number approximation.
Introductory Quantum Mechanics for Nanoscience
Published in Sunipa Roy, Chandan Kumar Ghosh, Chandan Kumar Sarkar, Nanotechnology, 2017
Chandan Kumar Ghosh, Sunipa Roy, Chandan Kumar Sarkar
Similar to classical mechanics, the situations where we can exactly apply quantum mechanics, that is, Schrödinger’s equation, are limited in number. To deal with real systems, several approximation methods have been developed. The developed methods are divided into time-independent and time-dependent depending on the time dependence of the Hamiltonian of the quantum system. In this section, we’ll discuss a few approximation methods, namely, the perturbation method, Wentzel-Kramers-Brillouin (WKB) approximation, and the variational method. The variational method relies on finding the wave function with the lowest energy level. Here, the other two approximation methods have been briefly described.
Bound state solutions, Fisher information measures, expectation values, and transmission coefficient of the Varshni potential
Published in Molecular Physics, 2021
E. Omugbe, O. E. Osafile, I. B. Okon, E. A. Enaibe, M. C. Onyeaju
The WKB approximation is an alternative analytical approach to obtain energy levels of physical systems without transforming the Schrodinger equation to a second-order differential equation which is characterised by some standard analytical methods [49–51]. The WKB approximation has been widely treated in [28]. In this present work, we will evaluate the energy levels and transmission coefficient starting from the respective quantisation condition and the transmission coefficient formula for the standard two turning point problem given as where are real classical turning points obtained from the equation . The classical momentum is obtained from Equation (10) as where is the effective potential given by