a particle, not just on what we can know about it with a certain
technique.
An estimate for electrons in atoms example 16
.A typical energy for an electron in an atom is on the order of
(1 volt)·e, which corresponds to a speed of about 1% of the speed
of light. If a typical atom has a size on the order of 0.1 nm, how
close are the electrons to the limit imposed by the uncertainty
principle?
.If we assume the electron moves in all directions with equal
probability, the uncertainty in its momentum is roughly twice its
typical momentum. This is only an order-of-magnitude estimate,
so we take∆pto be the same as a typical momentum:∆p∆x=pt y pi cal∆x
= (mel ect r on)(0.01c)(0.1× 10 −^9 m)
= 3× 10 −^34 J·sThis is on the same order of magnitude as Planck’s constant, so
evidently the electron is “right up against the wall.” (The fact that
it is somewhat less thanhis of no concern since this was only an
estimate, and we have not stated the uncertainty principle in its
most exact form.)
self-check FIf we were to apply the uncertainty principle to human-scale objects,
what would be the significance of the small numerical value of Planck’s
constant? .Answer, p. 1063
Discussion Questions
A Compare∆pand∆xfor the two lowest energy levels of the one-
dimensional particle in a box, and discuss how this relates to the uncer-
tainty principle.
B On a graph of∆pversus∆x, sketch the regions that are allowed and
forbidden by the Heisenberg uncertainty principle. Interpret the graph:
Where does an atom lie on it? An elephant? Can eitherporxbe mea-
sured with perfect accuracy if we don’t care about the other?13.3.5 Electrons in electric fields
So far the only electron wave patterns we’ve considered have
been simple sine waves, but whenever an electron finds itself in an
electric field, it must have a more complicated wave pattern. Let’s
consider the example of an electron being accelerated by the elec-
tron gun at the back of a TV tube. Newton’s laws are not useful,
because they implicitly assume that the path taken by the particle is
a meaningful concept. Conservation of energy is still valid in quan-
tum physics, however. In terms of energy, the electron is moving
from a region of low voltage into a region of higher voltage. Since902 Chapter 13 Quantum Physics