The term 2
ℜ
e
(
A
(
S
)
A
∗
(
T
)) in equation (1.12) is what generates quantum
interference mathematically. We shall see that in certain circumstances the
violations of equation (1.3) that are caused by quantum interference are not
detectable, so standard probability theory appears to be valid.
How do we know that the principle (1.11), which has these extraordinary
consequences, is true?
The soundest answer is that it is a fundamental
postulate of quantum mechanics, and that every time you look at a digital
watch, or touch a computer keyboard, or listen to a CD player, or interact
with any other electronic device that has been engineered with the help
of quantum mechanics, you are testing and vindicating this theory.
Our
civilisation now quite simply depends on the validity of equation (1.11).
6
Chapter 1: Probability and probability amplitudes
Figure 1.2
The probability distribu
tions of passing through each of the
two closely spaced slits overlap.
1.2.1 Twoslit interference
An imaginary experiment will clarify the physical implications of the prin
ciple and suggest how it might be tested experimentally.
The apparatus
consists of an electron gun, G, a screen with two narrow slits S
1
and S
2
,
and a photographic plate P, which darkens when hit by an electron (see
Figure 1.1).
When an electron is emitted by G, it has an amplitude to pass through
slit S
1
and then hit the screen at the point
x
. This amplitude will clearly
depend on the point
x
, so we label it
A
1
(
x
). Similarly, there is an amplitude
A
2
(
x
) that the electron passed through S
2
before reaching the screen at
x
.
Hence the probability that the electron arrives at
x
is
P
(
x
) =

A
1
(
x
) +
A
2
(
x
)

2
=

A
1
(
x
)

2
+

A
2
(
x
)

2
+ 2
ℜ
e
(
A
1
(
x
)
A
∗
2
(
x
))
.
(1
.
13)

A
1
(
x
)

2
is simply the probability that the electron reaches the plate after
passing through S
1
. We expect this to be a roughly Gaussian distribution
p
1
(
x
) that is centred on the value
x
1
of
x
at which a straight line from G
through the middle of S
1
hits the plate.

A
2
(
x
)

2
should similarly be a roughly
Gaussian function
p
2
(
x
) centred on the intersection at
x
2
of the screen and
the straight line from G through the middle of S
2
. It is convenient to write
A
i
=

A
i

e
i
φ
i
=
√
p
i
e
i
φ
i
, where
φ
i
is the phase of the complex number
A
i
.
Then equation (1.13) can be written
p
(
x
) =
p
1
(
x
) +
p
2
(
x
) +
I
(
x
)
,
(1
.
14a)
where the
interference term
I
is
I
(
x
) = 2
radicalbig
p
1
(
x
)
p
2
(
x
) cos(
φ
1
(
x
)
−
φ
2
(
x
))
.
(1
.
14b)
Consider the behaviour of
I
(
x
) near the point that is equidistant from the
slits. Then (see Figure 1.2)
p
1
≃
p
2
and the interference term is comparable
in magnitude to
p
1
+
p
2
, and, by equations (1.14), the probability of an
electron arriving at
x
will oscillate between
∼
2
p
1
and 0 depending on the
value of the phase difference
φ
1
(
x
)
−
φ
2
(
x
). In
§
2.3.4 we shall show that the
phases
φ
i
(
x
) are approximately linear functions of
x
, so after many electrons
have been fired from G to P in succession, the blackening of P at
x
, which
will be roughly proportional to the number of electrons that have arrived at
x
, will show a sinusoidal pattern.
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 Physics, mechanics, The Land, David Skinner, probability amplitudes, James Binney, Physics of Quantum Mechanics