416
B Simple Models
Fig. B.9 Hydrogen phase
space plot: p versus z for
F 0 =0
B.4.2 Nonzero Stark Field
Let us now assume that the system has dipole coupling to a constant electric field.
The Hamiltonian becomes
H o =
p 2
2μ
−
κ 0 e 2
z
− zF o = E o ,
(B.53)
where F 0 is proportional to the dipole moment and the electric field amplitude. The
momentum can be written in the form
p = ±
2μ
−|E o | +
κ 0 e 2
z
+ zF o .
(B.54)
The potential V (z) = −
κ 0 e 2
z − zF o is plotted in Fig. B.9. The constant field, F o ,
causes the potential to have a local maximum at z ∗ =
κ 0 e 2
F o
. Bound states will
occur for E o < 2
κ 0 e 2 F o and z < z ∗ . When E o < 2
κ 0 e 2 F o , there will be two
turning points, z ± , given by
z ± =
|E o |
2F o
⎡
⎣ 1 ±
1 −
4κ 0 e 2 F o
|E o | 2
⎤
⎦ .
(B.55)
The inner turning point, z − , is for a trajectory trapped in the well, while the outer
turning point, z + , is for a trajectory impinging on the potential barrier from the
region z > z ∗ .
B Simple Models
Fig. B.9 Hydrogen phase
space plot: p versus z for
F 0 =0
B.4.2 Nonzero Stark Field
Let us now assume that the system has dipole coupling to a constant electric field.
The Hamiltonian becomes
H o =
p 2
2μ
−
κ 0 e 2
z
− zF o = E o ,
(B.53)
where F 0 is proportional to the dipole moment and the electric field amplitude. The
momentum can be written in the form
p = ±
2μ
−|E o | +
κ 0 e 2
z
+ zF o .
(B.54)
The potential V (z) = −
κ 0 e 2
z − zF o is plotted in Fig. B.9. The constant field, F o ,
causes the potential to have a local maximum at z ∗ =
κ 0 e 2
F o
. Bound states will
occur for E o < 2
κ 0 e 2 F o and z < z ∗ . When E o < 2
κ 0 e 2 F o , there will be two
turning points, z ± , given by
z ± =
|E o |
2F o
⎡
⎣ 1 ±
1 −
4κ 0 e 2 F o
|E o | 2
⎤
⎦ .
(B.55)
The inner turning point, z − , is for a trajectory trapped in the well, while the outer
turning point, z + , is for a trajectory impinging on the potential barrier from the
region z > z ∗ .