pretty much fill the box, so our knowledge of the electron’s position
is of limited accuracy. If we write ∆xfor the range of uncertainty
in our knowledge of its position, then ∆xis roughly the same as the
length of the box:
∆x≈L
If we wish to know its position more accurately, we can certainly
squeeze it into a smaller space by reducingL, but this has an unin-
tended side-effect. A standing wave is really a superposition of two
traveling waves going in opposite directions. The equationp=h/λ
really only gives the magnitude of the momentum vector, not its
direction, so we should really interpret the wave as a 50/50 mixture
of a right-going wave with momentump=h/λand a left-going one
with momentump=−h/λ. The uncertainty in our knowledge of
the electron’s momentum is ∆p= 2h/λ, covering the range between
these two values. Even if we make sure the electron is in the ground
state, whose wavelengthλ= 2Lis the longest possible, we have an
uncertainty in momentum of ∆p=h/L. In general, we find
∆p&h/L,with equality for the ground state and inequality for the higher-
energy states. Thus if we reduceLto improve our knowledge of the
electron’s position, we do so at the cost of knowing less about its
momentum. This trade-off is neatly summarized by multiplying the
two equations to give
∆p∆x&h.
Although we have derived this in the special case of a particle in a
box, it is an example of a principle of more general validity:
The Heisenberg uncertainty principle
It is not possible, even in principle, to know the momentum and the
position of a particle simultaneously and with perfect accuracy. The
uncertainties in these two quantities are always such that ∆p∆x&
h.
(This approximation can be made into a strict inequality, ∆p∆x >
h/ 4 π, but only with more careful definitions, which we will not
bother with.)
Note that although I encouraged you to think of this deriva-
tion in terms of a specific real-world system, the quantum dot, no
reference was ever made to any specific laboratory equipment or pro-
cedures. The argument is simply that we cannotknowthe particle’s
position very accurately unless ithasa very well defined position,
it cannot have a very well defined position unless its wave-pattern
covers only a very small amount of space, and its wave-pattern can-
not be thus compressed without giving it a short wavelength and
a correspondingly uncertain momentum. The uncertainty princi-
ple is therefore a restriction on how much there is to know aboutSection 13.3 Matter as a wave 901