Problems 1

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Problems 1

• solutions

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Doppler again
y
?
x
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Some more mathematics

• Separation of variables

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Laplacian
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• Frequently in Physics we need to solve partial
  differential equations,
• e.g.

The Helmholtz Equation
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• Physical problems consist of a formal
  differential equation together with boundary
  conditions,

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• From the general theory of partial differential
  equations one can establish rules for when such
  an equation has a non-trivial solution(F 0)
• and if such a solution is unique.

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• You may assume unless otherwise stated that any
  p.d.e. you have to deal with has a solution and
  it is unique
• Thus it doesnt matter how you find your solution
  it just has to satisfy the diff eqn and the
  boundary conditions.

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Depends only on x
Depends only on y and z, independent of x
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Depends only on x
Depends only on y and z, independent of x
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Function only of z
Function only of y
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• We have turned our partial differential equation
  into sets of ordinary differential equations

Chosen to enforce boundary conditions
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Different coordinates

• Cartesian coordinates are good for boxes
• But we may be interested in problems with
  spherical symmetry or cyclindrical symmetry or
  other more complex symmetries
• The Laplacian can be transformed

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Cylindrical symmetry
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Spherical Symmetry
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• Rutherford discovered the nucleus
• Natural to think of atoms as little planetary
  systems

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• White light

Continuous Spectrum
White light
Emission Spectrum
Hot GAS
Absorption Spectrum
Cold Gas
White light
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The Balmer spectrum of hydrogen

• The Balmer series is in the visible
• the Lyman Series is in the UV

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The Bohr model

• Energy was quantized
• Angular Momentum was quantized

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Bohrs argument
Assume nucleus infinitely heavy Assume electron
is moving in circular orbit of radius,r,with
speed v then
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Bohr assumed that the orbital angular momentum
was quantized i.e. Consequently we obtain
fixed values of
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Ions and Exotic Atoms

• If we were dealing with hydrogenic ions
• That is to say a single electron and a nucleus
  of mass,M, charge,Z, then the analysis is
  essentially identical

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Exotic Atoms
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Atomic units
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• An attosecond is one quintillionth (10 -18 ) of a
  second
• Shortest laser pulse generated(80
  attosecond-(June 2008)
• http//technology.newscientist.com/channel/tech/dn
  14172-fastestever-flashgun-captures-image-of-light
  -wave.html?feedIdonline-news_rss20

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For Hydrogen Z1
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