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

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

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

4. The Quadratic Stark Effect Example
dropping
The Quadratic Stark Effect Example

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

11. 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.

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

13. 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

14. Example WKB tunneling
We can solve away from turning point

15. 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.

16. 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
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.