2 Mechanism of Population Inversion in N 2
+
27
field interaction:
H 0 ψ αvK m (r ) = ε αvK m ψ αvK m (r ),
(2.8)
and
H 0 = −
2
2μ
∂
2
∂r 2 + V α (r ) +
K (K + 1) − k
2
2μr 2
,
(2.9)
where k is the projection of the electronic orbital angular momentum on the molecular
axis connecting two nitrogen nuclei. To make use of the finite difference method, we
divide the space into a finite number of equally spaced nodes, separated by r . Thus
ψ αvK m (r ) becomes a vector with elements of its value at each point over the space:
ψ αvK m (r ) =
⎛
⎜
⎜
⎜
⎝
ψ αvK m (r 1 )
ψ αvK m (r 2 )
. . .
ψ αvK m (r max )
⎞
⎟
⎟
⎟
⎠
,
(2.10)
where r l = l · r . Since r stands for the internuclear distance of N 2
+
, r can only take
positive values, and is limited in the potential curve region of (0, r max ).
The second derivative of the kinetic part in H 0 is estimated from the first derivative.
Because the forward first-order derivative at r = r l is given by
∂
∂r
ψ αvK m | r =r l forward =
ψ αvK m (r l+1 ) − ψ αvK m (r l )
r
(2.11)
and the backward first-order derivative at r = r l is given by
∂
∂r
ψ αvK m | r =r l backward =
ψ αvK m (r l ) − ψ αvK m (r l+1 )
r
,
(2.12)
the second derivative can be expressed as
∂
2
∂r 2 ψ αvK m (r )| r =r l =
1
r
(
∂
∂r
ψ αvK m | r =r l forward −
∂
∂r
ψ αvK m | r =r l backward )
=
ψ αvK m (r l+1 ) − 2ψ αvK m (r l ) + ψ αvK m (r l−1 )
r 2
. (2.13)
In this way, the differential operator H 0 can be mapped to a difference operator.
By the substitution of (2.10) into (2.8) and the relationship shown in (2.13), the
Hamiltonian can be expressed as
H il =
⎧
⎪ ⎨
⎪ ⎩
−
1
2μ
−2
r 2 + V (r i ) +
K (K +1)−k
2
2μr i
2
when i = l,
−
1
2μ
1
r 2
when i − l = ±1,
0
when i = l and i − l = ±1.
(2.14)