Physics of wave packets

K.IshikawaHokkaido 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

２－１

Useful identity 　A

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

Widths in longitudinal and transverse directions are

２－３

x x

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

10 20

400

Experiment

wave packet

frequency

Attenuation length of UHCR

2,7[K]5.4[K]8.1[K]

10.8[K]

plane wave

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.