Principles of Quantum Mechanics   as Applied to Chemistry and Chemical Physics

Principles of Quantum Mechanics as Applied to Chemistry and Chemical Physics


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the probability density P(x) is postulated to be the square of the
absolute value of the wave function Ø(x)
P(x) ˆ jØ(x)j2 ˆ Ø\ufffd(x)Ø(x)
On the basis of this postulate, the interference pattern observed in the double-
slit experiment can be explained in terms of quantum particle behavior.
A particle, photon or electron, passing through slit A and striking the
detection screen at point x has wave function ØA(x), while a similar particle
passing through slit B has wave function ØB(x). Since a particle is observed to
retain its identity and not divide into smaller units, its wave function Ø(x) is
postulated to be the sum of the two possibilities
Ø(x) ˆ ØA(x)‡ØB(x) (1:48)
When only slit A is open, the particle emitted by the source S passes through
slit A, thereby causing the wave function Ø(x) in equation (1.48) to change or
collapse suddenly to ØA(x). The probability density PA(x) that the particle
strikes point x on the detection screen is, then
PA(x) ˆ jØA(x)j2
and the intensity distribution IA in Figure 1.9(a) is obtained. When only slit B
30 The wave function
is open, the particle passes through slit B and the wave function Ø(x) collapses
to ØB(x). The probability density PB(x) is then given by
PB(x) ˆ jØB(x)j2
and curve IB in Figure 1.9(a) is observed. If slit A is open and slit B closed
half of the time, and slit A is closed and slit B open the other half of the time,
then the resulting probability density on the detection screen is just
PA(x)‡ PB(x) ˆ jØA(x)j2 ‡ jØB(x)j2
giving the curve in Figure 1.9(b).
When both slits A and B are open at the same time, the interpretation
changes. In this case, the probability density PAB(x) is
PAB(x) ˆ jØA(x)‡ØB(x)j2
ˆ jØA(x)j2 ‡ jØB(x)j2 ‡Ø\ufffdA(x)ØB(x)‡Ø\ufffdB(x)ØA(x)
ˆ PA(x)‡ PB(x)‡ I AB(x) (1:49)
where
I AB(x) ˆ Ø\ufffdA(x)ØB(x)‡Ø\ufffdB(x)ØA(x)
The probability density PAB(x) has an interference term I AB(x) in addition to
the terms PA(x) and PB(x). This interference term is real and is positive for
some values of x, but negative for others. Thus, the term I AB(x) modifies the
sum PA(x)‡ PB(x) to give an intensity distribution with interference fringes as
shown in Figure 1.9(c).
For the experiment with both slits open and a detector placed at slit A, the
interaction between the wave function and the detector must be taken into
account. Any interaction between a particle and observing apparatus modifies
the wave function of the particle. In this case, the wave function has the form
of a wave packet which, according to equation (1.37), oscillates with time as
eÿiEt=". During the time period \u2dct that the particle and the detector are
interacting, the energy of the interacting system is uncertain by an amount \u2dcE,
which, according to the Heisenberg energy\u2013time uncertainty principle, equa-
tion (1.45), is related to \u2dct by \u2dcE > "=\u2dct. Thus, there is an uncertainty in the
phase Et=" of the wave function and ØA(x) is replaced by eijØA(x), where j
is real. The value of j varies with each particle\u2013detector interaction and is
totally unpredictable. Therefore, the wave function Ø(x) for a particle in this
experiment is
Ø(x) ˆ eijØA(x)‡ØB(x) (1:50)
and the resulting probability density Pj(x) is
1.8 Physical interpretation of the wave function 31
Pj(x) ˆ jØA(x)j2 ‡ jØB(x)j2 ‡ eÿijØ\ufffdA(x)ØB(x)‡ eijØ\ufffdB(x)ØA(x)
ˆ PA(x)‡ PB(x)‡ I j(x) (1:51)
where I j(x) is defined by
I j(x) ˆ eÿijØ\ufffdA(x)ØB(x)‡ eijØ\ufffdB(x)ØA(x)
The interaction with the detector at slit A has changed the interference term
from I AB(x) to I j(x).
For any particular particle leaving the source S and ultimately striking the
detection screen D, the value of j is determined by the interaction with the
detector at slit A. However, this value is not known and cannot be controlled;
for all practical purposes it is a randomly determined and unverifiable number.
The value of j does, however, influence the point x where the particle strikes
the detection screen. The pattern observed on the screen is the result of a large
number of impacts of particles, each with wave function Ø(x) in equation
(1.50), but with random values for j. In establishing this pattern, the term
I j(x) in equation (1.51) averages to zero. Thus, in this experiment the
probability density Pj(x) is just the sum of PA(x) and PB(x), giving the
intensity distribution shown in Figure 1.9(b).
In comparing the two experiments with both slits open, we see that interact-
ing with the system by placing a detector at slit A changes the wave function of
the system and the experimental outcome. This feature is an essential char-
acteristic of quantum theory. We also note that without a detector at slit A,
there are two indistinguishable ways for the particle to reach the detection
screen D and the two wave functions ØA(x) and ØB(x) are added together.
With a detector at slit A, the two paths are distinguishable and it is the
probability densities PA(x) and PB(x) that are added.
An analysis of the Stern\u2013Gerlach experiment also contributes to the
interpretation of the wave function. When an atom escapes from the high-
temperature oven, its magnetic moment is randomly oriented. Before this atom
interacts with the magnetic field, its wave function Ø is the weighted sum of
two possible states Æ and â
Ø ˆ cÆƇ cââ (1:52)
where cÆ and câ are constants and are related by
jcÆj2 ‡ jcâj2 ˆ 1
In the presence of the inhomogeneous magnetic field, the wave function Ø
collapses to either Æ or â with probabilities jcÆj2 and jcâj2, respectively. The
state Æ corresponds to the atomic magnetic moment being parallel to the
magnetic field gradient, the state â being antiparallel. Regardless of the
32 The wave function
orientation of the magnetic field gradient, vertical (up or down), horizontal (left
or right), or any angle in between, the wave function of the atom is always
given by equation (1.52) with Æ parallel and â antiparallel to the magnetic field
gradient. Since the atomic magnetic moments are initially randomly oriented,
half of the wave functions collapse to Æ and half to â.
In the Stern\u2013Gerlach experiment with two magnets having parallel magnetic
field gradients\u2013the \u2018first arrangement\u2019 described in Section 1.7\u2013all the atoms
entering the second magnet are in state Æ and therefore are all deflected in the
same direction by the second magnetic field gradient. Thus, it is clear that the
wave function Ø before any interaction is permanently changed by the inter-
action with the first magnet.
In the \u2018second arrangement\u2019 of the Stern\u2013Gerlach experiment, the atoms
emerging from the first magnet and entering the second magnet are all in the
same state, say Æ. (Recall that the other beam of atoms in state â is blocked.)
The wave function Æ may be regarded as the weighted sum of two states Æ9
and â9
Æ ˆ c9ÆÆ9‡ c9ââ9
where Æ9 and â9 refer to states with atomic magnetic moments parallel and
antiparallel, respectively, to the second magnetic field gradient and where c9Æ
and c9â are constants related by
jc9Æj2 ‡ jc9âj2 ˆ 1
In the \u2018second arrangement\u2019, the second magnetic field gradient is perpendicu-
lar to the first, so that
jc9Æj2 ˆ jc9âj2 ˆ 12
and
Æ ˆ 1
2
p (Æ9\ufffd â9)
The interaction of the atoms in state Æ with the second magnet collapses the
wave function Æ to either Æ9 or â9 with equal probabilities.
In the \u2018third arrangement\u2019, the right beam of atoms emerging from the
second magnet (all atoms being in state Æ9), passes through a third magnetic
field gradient parallel to the first. In this case, the wave function Æ9 may be
expressed as the sum of states Æ and â
Æ9 ˆ 1
2
p (Æ\ufffd â)
The interaction between the third magnetic field gradient and each atom
collapses the wave function Æ9 to either Æ or â with equal probabilities.
The interpretation of the various arrangements in the Stern\u2013Gerlach experi-
1.8 Physical