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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)
 
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