Physics of wave packets

1
Physics of wave packets

• K.Ishikawa
• Hokkaido University
• January 9 2009
• CollaboratorsTobita,Shimomura,
• Futamura,Hotta

2
Wave Packets

• 1 Wave packet Is a linear combination of plane
  waves and has a finite size of
• x and p.
• 2 Usually wave packet is well approximated by
  plane wave. In some places, wave packets give
  unique effects and are important. We study such
  phenomena.

3
Quantum vs classical mechanics

• Particles of the finite spatial extension are
• expressed by wave packets in QM and they
  resemble the particles of CM.
• However wave packets spread and sometimes
  spreading becomes large and they are approximated
  by plane waves and wave packet effects disappear
  then.
• When are wave packet effects important ?

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2 Uncertainty relation and spreading of wave
packet
Minimum wave packet of P_0 and X_0 is given by
In 3-dimensional space
Useful identity A
2-1
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Evolution of wave packet of the energy E(p) of
the mass m
In momentum representation,
Wave function in coordinate representation is
obtained by Fourier tr.
Is small, then it travels with group velocity
2-2
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Is large, then wave packet spreads
Stationary phase approximaton is applied,
Solution is
Integrate over p around px, then wave function
becomes
x
x
Widths in longitudinal and transverse directions
are
2-3
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Wave function
Normalizations maxima( use X1,P1 for X0,P0)
Position x in term of P0 is
At the neighbors of
2-4
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Wave function depends on the position ,x
Spreading of wave packet is described by,
Velocity of spreading
New uncertainty of spreading velocities
2-5
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Transverse d.
Longitudinal d.
time
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Spreading of wave packet

• Wave packet spreading is negligible in the
  longitudinal direction for the massless particle
• So wave packet effect is important
• in photon, neutrino and ultra-relativistic
• particles.
• Here we study background photons and solar
  photons.

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Cosmic background photon of 2.7K

• Experiment vs theory using wave packet

400
Experiment
wave packet
10
20
frequency
12
Attenuation length of UHCR
plane wave
2,7K
5.4K
8.1K
10.8K
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Solar Photons in visible energy region

• experiment

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Wave packet size of photon

• mean free path L,
• L1/(\sigma \rho)
• \sigmascattering cross section
• 1 Thomson scattering
• 2 Scattering with atom in ground state
  is small at certain wave length
• 3 Scattering with Rydberg atom is large
  and is enormous for very large n.

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1 QED correction in Planck distribution

• Corrections due to finite density and finite
  temperature ( Thomson scattering)
• Thermometer at high temperature

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3 Rydberg atom

• E-Ry 1/n2( large n)
• At large n, E is almost zero and the size of wave
  function becomes huge and infinity number of
  states around the zero energy do exist.
• At high temperature , where the rate of
  dissociation is between 0 and 1(by Sahas formula
  ), Rydberg atoms are expected to exist. They
  have large van der Waals force
• and large photon absorption rate.

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Conclusion

• Quantum field theory could have new
  applications .
• Quantum effects may be important even in Solar
  dynamics in outside region.
• Other applications of wave packets in LHC and
  others are possible.
• References
• K I and T.Shimomura,PTP
• K I and Y.Tobita, HEP , and in preparation
• K I , Y.Tobita,and Futamura (and KH) in
  preparation

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Winter school

• Many beautiful talks on lattice, string, field
  theory, gravity, BSM, QI and others
• Has been successful so far.
• free discussions
• Hope to be successful in the future too.