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)
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 magnetic
field adds harmonic
oscillator 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 spectrum
in 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)
This how two spectra
look
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 state
is 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 the
simplified field
does not exist
so the packet must
move with w in
CP field and you
get my result
(Actualy the only known
nondispersing
wavepacket in linear
quantum mechanics they always spread in
atoms)
But we also know that
Stark eigenstates really
do not exist
(they are rather
resonances since
Stark perturbation
series is usually
divergent for real
energy)
So adding true magnetic
field helps a bit (they now
don’t
ionize at all)