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)

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We all know Bohrresult for quantisation of
HydrogenEnergies are En-1/n2and states
withdifferent l and m are degenerate
3
External electric fieldchanges energy levelsof
Hydrogen(Stark effect)

• Hp2/2m-1/re E

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so the magnetic field(Zeeman effect)

• Hp2/2m-1/rw L_z

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True strong magneticfield adds
harmonicoscillator so would be great if it was
simpler

• Hp2/2m-1/r(w_c/2) L_z
• w_c2/8(x2y2)

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It can be generated by the rotation

• HUHU
• UeiwL_zt

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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 W1/n3 !!! (almost)
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This how two spectra look
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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)

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And you get the resultOne 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

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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)

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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 dont
• ionize at all)