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Mg
( )
=
exp(
gP Q
)
exp(
−
gP
)
−
Q
,
(11)
with the function
g
as a parameter; it is clear that
(12)
M
(0)
=
0.
From the double derivative of Eq. (11) with respect to
g
, we obtain
[
]
Mg
'
( )
=
exp( g
P P
)
, Q
exp(
−
g
P
)
(13.a)
[
]
'
∴
MP
(0)
=
,
Q
,
[
]
Mg
''
( )
=−
exp(
gP
)
⎡
P
,Q
,
P
⎤
exp(
−
gP
) ,
(13.b)
⎣
⎦
Then Eq. (2) can be substituted in Eq. (13.b) to derive:
''
() 2 ,
Mg
=
P
(14)
because the operators
exp (
commute with
P
. Now the integration of
±
g
P
)
Eq. (14) gives:
,
(15)
'
Mg
() 2
=
gP NPQ
+
( , )
[
]
Therefore,
; thus, Eq. (15) adopts the form
MN P
'
(0)
=
(
,
Q
)
(13.a)
P
,
Q
[
]
, whose integration implies that
'
() 2
Mg
=
gP
+
PQ
,
[
]
,
(16)
2
() g
Mg
=
P
+
g PQ
,
+
RPQ
( , )
But
R
(
P
,
Q
) = 0 due to Eq. (12). Then, Eqs. (11) and (16) give
[
]
2
exp(
gP Q
)
exp(
−
g P
)
−
Q
=
g P
+
g P Q
,
,
which is equivalent to first expression (3); in a similar manner, it is
possible to show the corresponding relations for
[ exp (
fQ
) , ]
P
and
[
]
fQ P Q
. It can be noted that the formula (8
.
105) given in the
reference [1] permit to give an alternative proof for Eq. (3).
[ exp (
) ,
,
]
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