Page 16 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics
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1.1 Wave motion 7
and the same minimum amplitude with the passage of each cycle. The
modulating function causes the maximum (or minimum) amplitude for each
cycle of the plane wave to oscillate with frequency Äù=2.
The pattern in Figure 1.2(a) propagates along the x-axis as time progresses.
After a short period of time Ät, the wave (1.8) moves to a position shown in
Figure 1.2(b). Thus, the position of maximum amplitude has moved in the
positive x-direction by an amount v g Ät, where v g is the group velocity of the
composite wave, and is given by
Äù
v g (1:9)
Äk
The expression (1.9) for the group velocity of a composite of two plane waves
is exact.
In the special case when k 2 equals ÿk 1 and ù 2 equals ù 1 in equation (1.7),
the superposition of the two plane waves becomes
Ø(x, t) e i(kxÿùt) e ÿi(kxùt) (1:10)
where
k k 1 ÿk 2
ù ù 1 ù 2
The two component plane waves in equation (1.10) travel with equal phase
velocities ù=k, but in opposite directions. Using equations (A.31) and (A.32),
we can express equation (1.10) in the form
Ø(x, t) (e ikx e ÿikx )e ÿiùt
2 cos kx e ÿiùt
2 cos kx (cos ùt ÿ i sin ùt)
We see that for this special case the composite wave is the product of two
functions: one only of the distance x and the other only of the time t. The
composite wave Ø(x, t) vanishes whenever cos kx is zero, i.e., when kx ð=2,
3ð=2, 5ð=2, ... , regardless of the value of t. Therefore, the nodes of Ø(x, t)
are independent of time. However, the amplitude or pro®le of the composite
wave changes with time. The real part of Ø(x, t) is shown in Figure 1.3. The
solid curve represents the wave when cos ùt is a maximum, the dotted curve
when cos ùt is a minimum, and the dashed curve when cos ùt has an
intermediate value. Thus, the wave does not travel, but pulsates, increasing and
decreasing in amplitude with frequency ù. The imaginary part of Ø(x, t)
behaves in the same way. A composite wave with this behavior is known as a
standing wave.
