Page 40 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

1.8 Physical interpretation of the wave function 31

is open, the particle passes through slit B and the wave function Ø(x) collapses
to Ø B (x). The probability density P B (x) is then given by
2
P B (x) ˆjØ B (x)j
and curve I B 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
2 2
P A (x) ‡ P B (x) ˆjØ A (x)j ‡jØ B (x)j
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 P AB (x)is
2
P AB (x) ˆjØ A (x) ‡ Ø B (x)j
2

2

ˆjØ A (x)j ‡jØ B (x)j ‡ Ø (x)Ø B (x) ‡ Ø (x)Ø A (x)
A
B
ˆ P A (x) ‡ P B (x) ‡ I AB (x) (1:49)
where

I AB (x) ˆ Ø (x)Ø B (x) ‡ Ø (x)Ø A (x)
A
B
The probability density P AB (x) has an interference term I AB (x) in addition to
the terms P A (x) and P B (x). This interference term is real and is positive for
some values of x, but negative for others. Thus, the term I AB (x) modi®es the
sum P A (x) ‡ P B (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 modi®es
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
e ÿiEt=" . During the time period Ät that the particle and the detector are
interacting, the energy of the interacting system is uncertain by an amount ÄE,
which, according to the Heisenberg energy±time uncertainty principle, equa-
tion (1.45), is related to Ät by ÄE >"=Ät. Thus, there is an uncertainty in the
ij
phase Et=" of the wave function and Ø A (x) is replaced by e Ø A (x), where j
is real. The value of j varies with each particle±detector interaction and is
totally unpredictable. Therefore, the wave function Ø(x) for a particle in this
experiment is
ij
Ø(x) ˆ e Ø A (x) ‡ Ø B (x) (1:50)
and the resulting probability density P j (x)is