1 / 12

Simple man theory of Trojan wave packets

Simple man theory of Trojan wave packets. (Larmor resonance in Hydrogen in electric field). We all know Bohr result for quantisation of Hydrogen Energies are En=-1/n^2 and states with different l and m are degenerate. External electric field changes energy levels of Hydrogen (Stark effect).

stesha
Download Presentation

Simple man theory of Trojan wave packets

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Simple man theory of Trojan wave packets (Larmor resonance in Hydrogen in electric field)

  2. We all know Bohrresult for quantisation of HydrogenEnergies are En=-1/n^2and states withdifferent l and m are degenerate

  3. External electric fieldchanges energy levelsof Hydrogen(Stark effect) H=p^2/2m-1/r+e E

  4. … so the magnetic field(Zeeman effect) H=p^2/2m-1/r+w L_z

  5. True strong magneticfield adds harmonicoscillator so would be great if it was simpler… H=p^2/2m-1/r+(w_c/2) L_z +w_c^2/8(x^2+y^2)

  6. It can be generated by the rotation H=U^+HU U=e^iwL_zt

  7. Now we consider Hydrogen spectrumin simplified magnetic field It clearly changes … For hydrogen alone states with different l and m have the same energy for the same n For hydrogen in simplified magnetic field states with the same DIFFERENCES N-l and l-m have if only W=1/n^3 !!! (almost)

  8. This how two spectra look

  9. Now add electric field and new Stark states can be found Apply perturbation calculus for new degeracy (diagonalize perturbation matrix in bases of denerate states (almost)

  10. And you get the result:One Stark stateis perfectly Gaussian (Trojan wavepacket) (You can actualy diagonalize the infinite matrix from Perturbation Theory with degeneracy noticing that it is almost tridiagonal and pendular

  11. But of course thesimplified field does not existso the packet mustmove with w in CP field and youget my result (Actualy the only known nondispersing wavepacket in linear quantum mechanics - they always spread in atoms)

  12. But we also know thatStark eigenstates reallydo not exist(they are rather resonances sinceStark perturbationseries is usually divergent for real energy) So adding true magnetic field helps a bit (they now don’t ionize at all)

More Related