TwoDimensional Scattering Across a TopHat Potential

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Two-Dimensional Scattering Across a Top-Hat
Potential

• Jennifer L. Ellsworth
• Wells College
• May 3, 2000

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Overview

• Define scattering
• Classical examples of scattering
• Scattering in the quantum realm
• Importance of scattering problems
• Top-hat potential
• Future directions

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Scattering Definition

• In general, scattering means to bring about a
  less orderly arrangement, either in position or
  direction.
• More specifically, the term denotes the change
  in direction of particles owing to collisions
  with other particles or systems.
• The system for this project was the top-hat
  potential.

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Transmission

• If a kangaroo jumps into a pool of water, he will
  be transmitted through the surface of the water
  into the pool.

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Scattering

• If the same kangaroo then tries to jump through
  concrete, he will bounce right back off again at
  some angle. This is scattering.

Concrete
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Quantum Effects

• Small particles like microscopic baseballs,
  protons, and electrons can behave differently.
  There is a positive probability that they will
  pass through barriers that classically are
  impenetrable.

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Tunneling

• This would be like our kangaroo jumping through a
  block of concrete.

Concrete
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Why is scattering important?

• Electricity and magnetism predict behavior of
  electromagnetic waves incident on charged
  particles.
• Particle physics predict interaction between
  microscopic particles. For example what happens
  when two protons collide?
• Nanoelectronics predict behavior of very, very
  small electronic components such as single
  electron transistors.

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Goal

• Predict the behavior of particles incident on a
  top-hat potential, when motion is constrained to
  two dimensions.

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Potential

• Potential is available energy, rather than active
  energy.
• A potential barrier is a region including a
  maximum of potential which hinders a low energy
  particle on one side of the region from passing
  to the other side.
• Example electrical repulsion of positively
  charged particles from the nucleus of an atom.

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Top-Hat Potential

• Top-hat potential energy curve of height V0 and
  radius a.
• The potential can be split into two regionsa
  region where there is zero potential and a region
  where there is potential V0.

a
V0
Zero potential
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Schrödingers Equation

• A particle can be described by a wave function.
• The wave function does NOT describe the physical
  path of the particle.
• The wave function DOES contain information about
  the probability a particle will be found in any
  given region of space.
• The wave function of a particle is an
  eigenfunction of Schrödingers equation.

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General Scattering Problem
Incident waves (red) and scattered waves (blue)
are shown relative to a scattering center or top
hat portion of the potential (yellow).
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Wave Functions

• The wave function is denoted by the symbol ?.
• Wave functions were found for the incident wave
  (shown in red in the last slide), and the general
  scattered wave (shown in blue in the last slide).

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Bessel Functions
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Cross Section and Phase Shift

• A measure of the probability of scattering a
  particle out of a beam.

• An angle that gives the relationship between the
  phase of the incident wave and the phase of the
  scattered wave.
• Value must be between -? and ?.

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Approach

• Approximate the ?inc at large r.
• Approximate the ?scatt at large r.
• Approximate the total wave function ?tot at large
  r.
• Set ?inc ?scatt ?tot then solve for cross
  section in terms of the phase shift.
• Evaluate results based on estimations of phase
  shift for long and short wavelengths.

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Asymptotic Wave Functions
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Details

• Set ?inc ?scatt ?tot
• Sines and cosines are linearly independent so the
  coefficients of each of these terms must be
  independently equal.
• This gives us two equations and two unknowns.
• Solve for Dl in terms of the phase angle.
• Solve for f(?) in terms of phase angle.
• Determine cross section in terms of phase angle.

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Result

• The total scattering cross section is

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Future Work

• Predict cross section in the limit of small and
  large wavelengths.
• How does the total cross section depend on the
  ratio E/V?
• Determine if there are any exact solutions to the
  problem.
• How does the total cross section change with time?

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Acknowledgements

• Dr. Steve Rae
• Dr. Carol Shilepsky
• Dr. Tom Stiadle
• Joan Poore

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Any Questions?