References - Bogoliubov Laboratory of Theoretical Physics - JINR

The Schroedinger equation is written as
i� dΨ
dt =[ˆ H + ˆ H ′ (t)]Ψ, (4)
where ˆ H is the time-independent Hamiltonian whose eigenfunctions satisfy the equations
ˆHψn = Wnψn. The exact wave function is written in the form
The coefficients cn(t) must satisfy the differential equations
Ψ= � cn(t)ψne −iWnt/� . (5)
i� dck(t)
dt = � H ′ kn (t)cn(t)e iωknt , (6)
where ωkn =(Wk − Wn)/�, andH ′ kn (t) =.
For the weak field transitions, one may use
ˆH ′ (t) =−μhσhxBx(t)sinωt − μJSJxBx(t)sinωt − μhσhzbz(t) − μJSJzbz(t), (7)
where bz(t) =(dBz/dx)vt.
The matrix elements are
H ′ 22 =(−μh − μJ)bz(t),
H ′ 24 =[−μh sin β − (μJ/ √ 2) cos β]Bx(t)sinωt,
H ′ 44 =[μh(sin 2 β − cos 2 β) − μJ sin 2 β]bz(t),
H ′ 45 =[−μh sin α cos β − μJ/ √ 2(sin α sin β +cosα cos β)]Bx(t)sinωt (8)
H ′ 55 =[μh(sin 2 α − cos 2 α)+μJ cos 2 α]bz(t),
H ′ 56 =[−μh cos α − (μJ/ √ 2) sin α]Bx(t)sinωt,
H ′ 66 =(μh + μJ)bz(t).
The values of sin α, cos α, sin β, cos β, ωik are taken at the initial value of Bz(xinit). To
find the final amplitude we must multiply the resulting wave function by Ψ ∗ n at the final
value of the field, Bz(xfinal).
Also, we need the matrix elements for the transition 1 − 3. We use the notations c1
and c3.
H ′ 11 =[μh(cos 2 β − sin 2 β) − μJ cos 2 β]bz(t),
H ′ 13 =[μh sin β cos α − μJ/ √ 2(sin α sin β +cosα cos β)]Bx(t)sinωt (9)
H ′ 33 =[μh(cos 2 α − sin 2 α)+μJ sin 2 α]bz(t),
We note that at a weak magnetic field (x ≪ 1) the level distance W1 − W3 ≈− 4
3 μJB,
and W2 − W4 = W4 − W5 = W5 − W6 ≈− 2
3 μJB. This is different from the case of
deuterium where all distances between the levels at F =3/2 andF =1/2 attheweak
magnetic field are equal to ≈− 2
3 μeB.
Let we have a system of two sextupoles with the space between them. Then, realizing
the transition 1 → 3 in the space between the sextupoles, after the second sextupole we
get the pure state 2 with F =3/2, mF =3/2 and after ionization in a strong magnetic
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