References - Bogoliubov Laboratory of Theoretical Physics - JINR

The energies of the states Ψ1 − Ψ6 are
W1 = ΔW
6
− μJ B
2
μJB − μhB, W3 = ΔW
ΔW
6
ΔW
2
B − μJ
2
− ΔW
2
6
+ ΔW
2
+ μJ B
2
�
1+ 2
+ ΔW
2
3 x + x2 , W2 = − ΔW
�
1+ 2
3x + x2 , W5 = ΔW
6
�
1 − 2
3 x + x2 , W6 = − ΔW
3 + μJB + μhB.
3 −
�
1 − 2
3x + x2 , W4 =
B + μJ 2 −
2 The Schroedinger equation in
the uncoupled state basis
W/ W
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
1
2
3
4
5
0 0.5 1 1.5 2
B/Bc
Figure 1: The scheme of
3 He 2 3 S1 hyperfine structure
and Zeeman splitting, Bc =
0.2407 T.
The adiabatic transitions for the hyperfine states of hydrogen
were considered by Antishev and Belov [1]. Oh [2] published detailed results for the
weak field transitions (WFT) in deuterium. Here we solve this problem for 3 He in the
metastable state 2 3 S1.
In the uncoupled |mh,mJ > state basis
Ψ(t) =C1(t)ψ +
h ψ+
+
J +C2(t)ψ h ψ0 +
J +C3(t)ψ h ψ−
J
+C4(t)ψ −
and we obtain the following equations for the amplitudes:
h ψ+
−
J +C5(t)ψh ψ0 −
J +C6(t)ψh ψ−
J
6
, (2)
dC1/dt = −i/�{C1[−(μh + μJ)Bz +ΔW/3] − C2μJBx/ √ 2 − C4μhBx}
dC2/dt = −i/�{−C1μJBx/ √ 2 − C2μhBz − C3μJBx/ √ √
2+C4 2ΔW/3 − C5μhBx}.
dC3/dt = −i/�{−C2μJBx/ √ √
2+C3[(−μh+μJ)Bz−ΔW/3]+C5 2ΔW/3−C6μhBx}. (3)
√
dC4/dt = −i/�{−C1μhBx + C2 2ΔW/3+C4[(μh − μJ)Bz − ΔW/3] − C5μJBx/ √ 2}.
√
dC5/dt = −i/�{−C2μhBx + C3 2ΔW/3 − C4μJBx/ √ 2+C5μhBz − C6μJBx/ √ 2}.
dC6/dt = −i/�{−C3μhBx − C5μJBx/ √ 2+C6[(μh + μJ)Bz +ΔW/3]}.
3 The Schroedinger equation in the basis of the stationary
states
Another way is to solve the Schroedinger equation in the basis of the stationary states,
as it was made by Beijers [3] for hydrogen. But he used the ”static” Hamiltonian slowly
changing with time because atoms move through a changing magnetic field. Hasuyama
and Wakuta [4] used the ”static” Hamiltonian for strong field transitions in deuterium.
This approach is not always correct.
We use, as the basis, the stationary states existing at the value of the magnetic field,
Bz = B0, at the enter into the region of a RF field.
For WFT 2–6 we consider only the four-level system of 2,4,5 and 6 substates of the
F =3/2 state, since the levels of the F =1/2 state are sufficiently distant as to have no
significant effect in our problem. For the amplitudes of these states, we use the notations
c2 − c6.
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