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p
j
N
1
2
m
j
+
κ
j
q
j
(
,
)
=
.
H
p
q
(3.28)
j
=
1
We now consider the transformation variables for each oscillator in the form
√
2
q
j
+
ic
jP
j
,
1
a
j
=
(3.29)
√
2
q
j
−
ic
jP
j
,
1
a
j
=
where
c
j
is a constant to be determined. Substituting these new variables into (
3.28
)
yields the new Hamiltonian
m
j
ω
a
j
+
2
m
j
ω
a
j
a
j
a
∗
j
N
1
m
j
c
j
1
m
j
c
j
H
(
a
∗
)
=
2
2
a
,
c
j
j
−
+
j
+
j
=
1
(3.30)
and selecting the constants
c
j
=
1
/(
m
j
ω
j
)
reduces the new Hamiltonian to the form
N
H
(
a
∗
)
=
1
ω
j
a
j
a
j
.
a
,
(3.31)
j
=
The transformation (
3.29
) is canonical because it preserves Hamilton's equations in the
form
H
da
j
dt
=−
i
∂
a
j
,
∂
da
j
dt
=
H
i
∂
(3.32)
∂
a
j
and the variable
a
j
is independent of its complex conjugate
a
j
.
The equation of motion
for each oscillator in the case of
N
independent oscillators is given by
da
j
dt
=−
i
ω
j
a
j
,
(3.33)
whose solution is
e
−
i
ω
j
t
a
j
(
t
)
=
a
j
(
0
)
(3.34)
together with its complex conjugate.
Here we have reduced the linear harmonic oscillator to its essential characteris-
tics. It is a process with a constant amplitude and initial phase specified by the initial
displacement
a
j
(
,
A
j
≡
0
)
(3.35)
Im
a
j
(
)
0
φ
j
≡−
(3.36)
Re
a
j
(
0
)
and a time-dependent phase
(
ω
j
t
+
φ
j
)
,
A
j
e
−
i
a
j
(
t
)
=
(3.37)