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## Simple man theory of Trojan wave packets

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**Simple man theory of Trojan wave packets**(Larmor resonance in Hydrogen in electric field)**We all know Bohrresult for quantisation of HydrogenEnergies**are En=-1/n^2and states withdifferent l and m are degenerate**External electric fieldchanges energy levelsof**Hydrogen(Stark effect) H=p^2/2m-1/r+e E**… so the magnetic field(Zeeman effect)**H=p^2/2m-1/r+w L_z**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)**It can be generated by the rotation**H=U^+HU U=e^iwL_zt**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)**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)**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**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)**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)