Loading...

PPT – Quantum Mechanics for Applied Physics PowerPoint presentation | free to view - id: c1b68-YmUyY

The Adobe Flash plugin is needed to view this content

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

dropping

The Quadratic Stark Effect Example

(No Transcript)

Optical Phased Array Multiple Parallel Optical

Waveguides

Output Fibers

GaAs Waveguides

WG 1

WG 128

Air Gap

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

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.

We need an asymptotic expansion of the solutions

of the Schrödinger equation in h.

We expanded F(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/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.

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.