Information Technology Reference
In-Depth Information
The formal solution to the linear equation of motion (
3.10
) is given by
q
(
t
)
=
A
0
cos
[
ω
0
t
+
φ
0
]
,
(3.12)
with the amplitude
A
0
and phase
φ
0
determined by the oscillator's initial conditions
p
2
(
0
)
q
2
A
=
(
0
)
+
0
,
(3.13)
2
ω
p
(
0
)
φ
0
=−
)
.
tan
(3.14)
ω
0
q
(
0
Figure
3.1
(a) depicts the harmonic-oscillator potential with two distinct energy levels
shown as flat horizontal line segments. Figure
3.1
(b)isthe
(
,
)
phase space for the
oscillator and the two curves are energy-level sets, namely curves of constant energy for
the two energy levels in Figure
3.1
(a). The energy-level sets are ellipses for
p
q
ω
0
>
1 and
circles for
ω
0
=
1
.
(a)
1
0.8
0.6
0.4
0.2
-75
-50
-25
Displacement
q
0
25
50
75
100
(b)
1
0.75
0.5
0.25
0
-0.25
-0.5
-0.75
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
Displacement
q
Figure 3.1.
(a) The harmonic potential function
V
(
q
)
is plotted versus
q
and two energy levels are denoted.
(b) The orbits for the energy level depicted in (a) are shown in the
(
q
,
p
)
phase space.
