Physics of wave packets

1. Physics of wave packets K.Ishikawa Hokkaido University January 9 2009 Collaborators:Tobita,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　Usuallywave 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 ?

4. 2 Uncertainty relationand spreading of wave packet Minimum wave packet of P_0 and X_0 is given by In 3-dimensional space Useful identityA ２－１

5. 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

6. 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

7. Wave function Normalization’s maxima( use X1,P1 for X0,P0) Position x in term of P0 is At the neighbors of ２－４

8. Wave function depends on the position ,x Spreading of wave packet is described by, Velocity of spreading New uncertainty of spreading velocities ２－５

9. Transverse d. Longitudinal d. time

10. 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.

11. 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 2,7[K] 5.4[K] 8.1[K] 10.8[K]

13. Solar Photons in visible energy region • experiment

14. 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.

15. 1 QED correction in Planck distribution • Corrections due to finite density and finite temperature ( =Thomson scattering) • Thermometer at high temperature

16. 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.

17. 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

18. Winter school • Many beautiful talks on lattice, string, field theory, gravity, BSM, QIand others • Has been successful so far. “ free discussions ” Hope to be successful in the future too.