Download Physics of wave packets

Physics of wave packets
K.Ishikawa
Hokkaido University
January 9 2009
Collaborators:Tobita,Shimomura,
Futamura,Hotta
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.
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 ?
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
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
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
Wave function
Normalization’s maxima( use X1,P1 for X0,P0)
Position x in term of P0 is
At the neighbors of
２－４
Wave function depends on the position ,x
Spreading of wave packet is described by,
Velocity of spreading
New uncertainty of spreading velocities
２－５
Transverse d.
Longitudinal d.
time
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.
Cosmic background photon of 2.7K
• Experiment vs theory using wave packet
400
Experiment
wave packet
10
frequency
20
Attenuation length of UHCR
plane wave
2,7[K]
5.4[K]
8.1[K]
10.8[K]
Solar Photons in visible energy
region
• experiment
Wave packet size of photon
• mean free path L,
L=1/(\sigma \rho)
• \sigma=scattering 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.
1 QED correction in Planck
distribution
• Corrections due to finite density and finite
temperature ( =Thomson scattering)
• Thermometer at high temperature
3 Rydberg atom
• E=-Ry 1/n^2( 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 Saha’s
formula ), Rydberg atoms are expected to exist.
They have large van der Waals force
and large photon absorption rate.
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
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.