Quantum Mechanics for Applied Physics

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Quantum Mechanics for Applied Physics

• Lecture IV
• Perturbation theory
• Time independent
• Time dependent
• WKB

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

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Time independent perturbation

• The zeroth-order term
• First Order

. Taking the inner product of both sides with
using
General
using
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dropping
The Quadratic Stark Effect Example
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Optical Phased Array Multiple Parallel Optical
Waveguides
Output Fibers
GaAs Waveguides
WG 1
WG 128
Air Gap
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WKB (WentzelKramersBrillouin) approximation
ID Schrödinger equation
Smooth varying potential over long scale larger
then local wavelength
When h ?0, ? ?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.
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We need an asymptotic expansion of the solutions
of the Schrödinger equation in h.
We expanded F(x) in powers of h
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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
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Example WKB tunneling
We can solve away from turning point
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Turning points

• The exact solution Ai(s), sketched at the right
  in the neighborhood of the turning point, has the
  asymptotic form (1/p1/2s1/4)cos(2/3)s3/2 - p/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 p/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 p/2
from the two ends to allow for the connection.
This gives us S (n 1/2)p. The phase integral
for a harmonic oscillator with energy W is S
Wp/h? , so we find W (n 1/2)h?. 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.
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Application of the WKB Approximation in the
Solution of the Schrödinger Equation
Zbigniew L. Gasyna and John C. LightDepartment
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.