Page 27 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics
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18 The wave function
As time increases from ÿ1 to 0, the half width of the wave packet jØ ã (x, t)j
continuously decreases and the maximum amplitude continuously increases. At
p
t 0 the half width attains its lowest value of 2=á and the maximum
p
amplitude attains its highest value of 1= 2ð, and both values are in agreement
with the wave packet in equation (1.20). As time increases from 0 to 1, the
half width continuously increases and the maximum amplitude continuously
2
decreases. Thus, as t increases, the wave packet jØ ã (x, t)j remains gaussian
in shape, but broadens and ¯attens out in such a way that the area under the
2
square jØ ã (x, t)j of the wave packet remains constant over time at a value of
p
ÿ1
(2 ðá) , in agreement with Parseval's theorem (1.18).
The product ÄxÄk for this spreading wave packet Ø ã (x, t)is
p
4 2 2
ÄxÄk 2 1 4á ã t
and increases as jtj increases. Thus, the value of the right-hand side when t 0
is the lower bound for the product ÄxÄk and is in agreement with the
uncertainty relation (1.23).
1.4 Particles and waves
To explain the photoelectric effect, Einstein (1905) postulated that light, or
electromagnetic radiation, consists of a beam of particles, each of which travels
at the same velocity c (the speed of light), where c has the value
8
c 2:997 92 3 10 ms ÿ1
Each particle, later named a photon, has a characteristic frequency í and an
energy hí, where h is Planck's constant with the value
h 6:626 08 3 10 ÿ34 Js
The constant h and the hypothesis that energy is quantized in integral multiples
of hí had previously been introduced by M. Planck (1900) in his study of
1
blackbody radiation. In terms of the angular frequency ù de®ned in equation
(1.2), the energy E of a photon is
E "ù (1:32)
where " is de®ned by
h ÿ34
" 1:054 57 3 10 Js
2ð
Because the photon travels with velocity c, its motion is governed by relativity
1 The history of the development of quantum concepts to explain observed physical phenomena, which
occurred mainly in the ®rst three decades of the twentieth century, is discussed in introductory texts on
physical chemistry and on atomic physics. A much more detailed account is given in M. Jammer (1966)
The Conceptual Development of Quantum Mechanics (McGraw-Hill, New York).
