Approximation methods in Quantum Mechanics

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Approximation methods in Quantum Mechanics
Kap. 7-lect2
Introduction to
Time dependent
Time-independent methods
Methods to obtain an approximate eigen energy, E
and wave function
Golden Rule
perturbation methods
Methods to obtain an approximate expression for
the expansion amplitudes.
Ground/Bound states
Continuum states
Perturbation theory
Variational method
Scattering theory
Non degenerate states
Degenerate states
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Scattering Theory

• Classical Scattering
• Differential and total cross section
• Examples Hard sphere and Coulomb scattering

• Quantal Scattering
• Formulated as a stationary problem
• Integral Equation
• Born Approximation
• Examples Hard sphere and Coulomb scattering

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The Scattering Cross Section (To be corrected,
see Endre Slide)
per unit time
Differential Cross Section
Total Cross Section
Dimension AreaInterpretation Effective area
for scattering.
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The Scattering Cross Section
Differential Cross Section
Total Cross Section
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Quantal Scattering - No Trajectory! (A plane
wave hits some object and a spherical wave
emerges)
Procedure

• Solve the time independent Schrödinger equation
• Approximate the solution to one which is valid
  far away from the scattering center
• Write the solution as a sum of an incoming plane
  wave and an outgoing spherical wave.
• Must find a relation between the wavefunction and
  the current densities that defines the cross
  section.

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Current Density
Incomming current density
Outgoing spherical current density
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Example - Classical scattering
Hard Sphere scattering
Geometrical Cross sectional area of sphere!
Independent of angles!
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Example from 1D
In this case (since potential is discontinuous)
we can find f excactly by gluing
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The Schrödinger equation - scattering form
Now we must define the current densities from the
wave function
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The final expression
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Summary

Then we have
. Now we can start to work
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Integral equation
With the rewritten Schrödinger equation we can
introducea Greens function, which (almost)
solves the problem for a delta-function
potential
Then a solution of
can be written
where we require
because.
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This term is 0
This equals
Integration over the delta function gives result
Formal solution
Useless so far!
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Must find G(r) in
Note
Then
The function
solves the problem!
Proof
The integral can be evaluated, and the result is
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implies that
Inserting G(r), we obtain
At large r this can be recast to an outgoing
spherical wave..
The Born series
And so on. Not necesarily convergent!
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SUMMARY
We obtains
At large r this can be recast to an outgoing
spherical wave..
The Born series
And so on. Not necesarily convergent!
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Asymptotics - Detector is at near infinite r
The potential is assumed to have short range,
i.e. Active only for small r
1)
2)
Asymptotic excact result
Still Useless!
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The Born approximation
The scattering amplitude is then
) The momentum change Fourier transform of the
potential!
Valid when
Weak potentials and/or large energies!
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Spheric Symmetric potentials
Total Cross Section
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Summary - 1st. Born Approximation
Best at large energies!
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Example - Hard sphere 1. Born scattering
Classical Hard Sphere scattering
Quantal Hard Sphere potential
Thats it!
Depends on angles - but roughly independent when
qR ltlt 1