Quantum Mechanics for Applied Physics Lecture IV Perturbation theory Time independent Time dependent WKB

Time-Independent Perturbation Theory We begin with Small perturbation Choosing a parameter expanding and

Time independent perturbation The zeroth-order term First Order . Taking the inner product of both sides with using General using

The Quadratic Stark Effect Example dropping The Quadratic Stark Effect Example

Optical Phased Array – Multiple Parallel Optical Waveguides Output Fibers GaAs Waveguides WG #1 WG #128 Air Gap Input Optical Fiber

WKB (Wentzel–Kramers–Brillouin) approximation ID Schrödinger equation: Smooth varying potential over long scale larger then local wavelength When h 0, λ 0 and the potential is always smooth Local momentum The approximate wavefunction can then be written in terms of the phase accumulated from x0 to x as where +/- corresponds to the right/left moving wave.

We need an asymptotic expansion of the solutions of the Schrödinger equation in h. We expanded Φ(x) in powers of h

WKB semi classical Classical probability to find a free particle: The approximation breaks when: The approximation breaks at turning points need smooth long potential Exactly as WKB

Example WKB tunneling We can solve away from turning point

Turning points The exact solution Ai(s), sketched at the right in the neighborhood of the turning point, has the asymptotic form (1/π1/2s1/4)cos[(2/3)s3/2 - π/4] to the right of the turning point. To the left, it decreases exponentially as required. The net effect is that the WKB approximate solution is pushed away from the turning point by an eighth of a wavelength, or phase π/4, in the asymptotic region. The Airy function can be expressed in terms of Bessel functions of order 1/3. Therefore, we can carry the phase integral from turning point to turning point, as in the case of the infinite square well, and subtract the π/2 from the two ends to allow for the connection. This gives us S = (n + 1/2)π. The phase integral for a harmonic oscillator with energy W is S = Wπ/hν , so we find W = (n + 1/2)hν. Surprisingly, this is the exact result, in spite of the fact that our method is approximate. The connection relations supply the 1/2 that implies a zero-point energy, which is not present in the old quantum theory.

Application of the WKB Approximation in the Solution of the Schrödinger Equation Zbigniew L. Gasyna and John C. Light Department of Chemistry, The University of Chicago, Chicago, IL 60637-1403 computational experiment is proposed in which the WKB approximation is applied in the solution of the Schrödinger equation. Energy levels of bound states are calculated for a diatomic oscillator for which the potential energy is defined by a simple function, such as the Morse or Lennard-Jones potential. Application of the WKB method to calculating the group velocities and attenuation coefficients of normal waves in the arctic underwater waveguide Krupin V. D. ; 2005 Algorithms based on the WKB approximation are proposed for the fast and accurate calculation of the group time delays and effective attenuation coefficients of normal waves in the deep-water sound channel of the Arctic Ocean. These characteristics of the modes are determined in the adiabatic approximation.