Page 49 - Electrical Properties of Materials

32 The electron as a wave

which can be expressed in terms of wavelength as
h
λ = , (2.32)
mv g
and this is nothing else but de Broglie’s relationship. Thus, if we assume the
validity of the wave picture, identify the group velocity of a wave packet with
the velocity of an electron, and assume that the centre frequency of the wave
packet is related to the energy of the electron by Planck’s constant, de Broglie’s
relationship automatically drops out.
This proves, of course, nothing. There are too many assumptions, too many
identifications, representations, and interpretations; but, undeniably, the differ-
ent pieces of the jigsaw puzzle do show some tendency to fit together. We have
now established some connection between the wave and particle aspects, which
seemed to be entirely distinct not long ago.
What can we say about the electron’s position? Well, we identified the posi-
tion of the electron with the position of the wave packet. So, wherever the wave
packet is, there is the electron. But remember, the wave packet is not infinitely
narrow; it has a width z, and there will thus be some uncertainty about the
If we know the position of the position of the electron.
electron with great precision, that Let us look again at eqn (2.20). Taking note of the relationship expressed in
is if z is very small, then the eqn (2.31) between wavenumber and momentum, eqn (2.20) may be written as
uncertainty in the velocity of the
electron must be large. p z = h. (2.33)
This is known as Heisenberg’s (Nobel Prize, 1932) uncertainty relationship.
It means that the uncertainty in the position of the electron is related to the
uncertainty in the momentum of the electron. Let us put in a few figures to see
the orders of magnitude involved. If we know the position of the electron with
an accuracy of 10 –9 m then the uncertainty in momentum is
–1
p =6.6 × 10 –25 kg m s , (2.34)
corresponding to
~ 5 –1
v = 7 × 10 ms , (2.35)
that is, the uncertainty in velocity is quite appreciable.
Taking macroscopic dimensions, say 10 –3 m for the uncertainty in position,
and a bullet with a mass 10 –3 kg, the uncertainty in velocity decreases to
–1
v =6.6 × 10 –28 ms , (2.36)
which is something we can easily put up with in practice. Thus, whenever we
come to very small distances and very light particles, the uncertainty in ve-
locity becomes appreciable, but with macroscopic objects and macroscopic
distances the uncertainty in velocity is negligible. You can see that everything
here depends on the value of h, which happens to be rather small in our
40
universe. If it were larger by a factor of, say 10 , the police would have
considerable difficulty in enforcing the speed limit.
The uncertainty relationship has some fundamental importance. It did away
(probably for ever) with the notion that distance and velocity can be simultan-
eously measured with arbitrary accuracy. It is applicable not only to position