
April 25, 2014 11:1 PSP Book - 9in x 6in 06-Rashid-A.-Ganeev-c06
196 Characterization of Plasma Harmonics
It is convenient to introduce the following transformation matrix
V
nm
(
t
)
=
u
∗
n
(
r
)
exp
−i
q
c
χ
(
r, t
)
u
m
(
r
)
dV. (6.8)
As we have mentioned above, in the case
χ
(
r, t
)
= A
(
t
)
r, (6.9)
the Hamiltonian of the problem (6.4) takes the form
H =
1
2m
p −
q
c
A
(
t
)
2
+U
(
r
)
.
and coincides with the Hamiltonian of Eq. (6.1). Hence, the
eigenfunctons of the Hamiltonian of the TDSE are
ϕ
n
(
r, t
)
= u
n
(
r
)
exp
i
q
c
A
(
t
)
r
. (6.10)
One can see that Eq. (6.10) looks similar to the formula for
gauge transformation proposed by Goppert–Mayer [58]. However,
it should be noted here that this