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26 The electron as a wave
Screen Wave
with holes detector
P
1
P = P + P
12 1 2
A
Source
of •
waves B
P 2
Fig. 2.2
An experiment with waves.
against distance from the gun nozzle–slit A axis. Calling this pattern P 1 ,we
expect (and get) a similar but displaced pattern P 2 if slit B is opened and slit A
is closed. Now if both slits are open, the combined pattern, P 12 ,issimply
P 12 = P 1 + P 2 . (2.1)
We will now think of a less dangerous and more familiar experiment, with
waves in a ripple tank (Fig. 2.2). The gun is replaced by a vibrator or ripple-
generator, the slits are the same, and in the target plane there is a device to
measure the ripple intensity, that is a quantity proportional to the square of the
height of the waves produced. Then, with one slit open we find
2
P 1 =|h 1 | , (2.2)
or with the other open
2
P 2 =|h 2 | , (2.3)
where we have taken h 1 and h 2 as complex vectors and the constant of propor-
tionality is unity. The probability functions are similar to those obtained with
one slit and bullets. So far, waves and bullets show remarkable similarity. But
with both slits open we find
P 12 = P 1 + P 2 . (2.4)
Instead, as we might intuitively suppose, the instantaneous values of the wave
heights from each slit add; and as the wavelength of each set of ripples is the
same, they add up in the familiar way
δ is the phase difference between 2 2 2
P 12 =|h 1 + h 2 | =|h 1 | +|h 2 | +2|h 1 ||h 2 | cos δ. (2.5)
the two interfering waves.
Thus, the crucial difference between waves and particles is that waves interfere,
but particles do not.
The big question in the 1920s was: are electrons like bullets or do they
∗
∗ Nobel Prize, 1929. follow a theoretical prediction by L. de Broglie ? According to de Broglie,
