(b) Higher-order WKB corrections: Using WKB approximation method, the lowest order of the equation of motion describing a particle moving in the black hole gives the Hamilton-Jacobi equation.
Using the WKB approximation the tunneling probability for the classically forbidden trajectory through the horizon is given by
Another scenario where f(R) theory has been considered is the origin of the universe by a tunneling mechanism from "nothing" to the de Sitter phase of Starobinski's model [8], where a description of the universe in the framework of quantum cosmology is given, from which the tunneling probability, the subsequent curvature fluctuations, and the duration of the inflationary phase were computed, in the
WKB approximation. Quantum cosmology of f(R) theories has been studied also in [9-12].
Various methods are employed to solve singular perturbation problems analytically, numerically, or asymptotically such as the method of matched asymptotic expansions (MMAE), the method of multiple scales, the method of
WKB approximation, Poincare-Lindstedt method and periodic averaging method.
Using the
WKB approximation in inhomogeneous plasma the starting point is a solution of the form
If amplitudes or phases of the solutions can be assumed as slowly varying quantities, asymptotic approaches such as the
WKB approximation can be applied to find approximate wave solutions [12-16].
The semiclassical tunneling method by using the Hamilton-Jacobi ansatz with
WKB approximation is another way to obtain the Bekenstein-Hawking temperature and the tunneling rate as [GAMMA] [approximately equal to] exp[-2fmS] [19].
Using the
WKB approximation, the action can be chosen at the leading order in [??] as
Then, they used the
WKB approximation to show that the solution to the deformed Schrodinger equation:
(iii) We employ the
WKB approximation [45, 46] to evaluate the quasinormal frequencies.
Most famous derivation is made by Parikh and Wilczek in which they use a tunnelling process with a semiclassical
WKB approximation [4-12].