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32 The wave function
2
2
P j (x) jØ A (x)j jØ B (x)j e ÿij ij
Ø (x)Ø B (x) e Ø (x)Ø A (x)
A B
P A (x) P B (x) I j (x) (1:51)
where I j (x) is de®ned by
Ø (x)Ø B (x) e Ø (x)Ø A (x)
I j (x) e ÿij ij
B
A
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 unveri®able number.
The value of j does, however, in¯uence 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 P j (x) is just the sum of P A (x) and P B (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 Ø A (x) and Ø B (x) are added together.
With a detector at slit A, the two paths are distinguishable and it is the
probability densities P A (x) and P B (x) that are added.
An analysis of the Stern±Gerlach 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 ®eld, its wave function Ø is the weighted sum of
two possible states á and â
Ø c á á c â â (1:52)
where c á and c â are constants and are related by
2
2
jc á j jc â j 1
In the presence of the inhomogeneous magnetic ®eld, the wave function Ø
2
2
collapses to either á or â with probabilities jc á j and jc â j , respectively. The
state á corresponds to the atomic magnetic moment being parallel to the
magnetic ®eld gradient, the state â being antiparallel. Regardless of the
