x (1000 kg)(1 m/s) 6.626
2
21
0
(^34) J s
where all of the values of the variables have been substituted into equation
10.5. Solving for x:
x5.27 10 ^38 m
You may want to verify not just the numbers but how the units work out.
This minimum uncertainty is undetectable even using modern measure-
ments of position and so this lower limit on the measurement would never
be noticed.
b.For a small electron, using the same equation but different numbers:
x(9.109 10 ^31 kg)(2.00 104 m/s)
6.626
2
2
1
0
(^34) J s
where we have used the mass of the electron and, for 1% of the velocity of
the electron, [0.01(2.00 106 ) 2.00 104 ]. Solving for x:
x2.89 10 ^9 m 2.89 nm 28.9 Å
The uncertainty in the position of the electron is at least 3 nanometers, sev-
eral times larger than atoms themselves. It would be easy to notice experi-
mentally that one couldn’t pin down the position of an electron to within
3 nm!
The above example illustrates that the idea of uncertainty cannot be ignored
at the atomic level.Certainly, if the velocity were known to lower precision, say,
to one part in ten, the corresponding minimum uncertainty in the position
would be lower. But the uncertainty principle states mathematically that as one
goes up, the other goes down, and neither can be zero for simultaneous deter-
minations. The uncertainty principle does not address a maximum uncer-
tainty, so the uncertainty can be (and usually is) larger. But some measure-
ments have a fundamental limit to how exactly they can be determined
simultaneously with other observables.
Finally, position and momentum are not the only two observables whose
uncertainties are related through an uncertainty principle. (In fact, another
mathematical form of the uncertainty principle is expressed in terms of the op-
erators for the observables, like equations 10.3 and 10.4, and not the observ-
able values themselves.) There are many such combinations of observables, like
multiple components of angular momentum. There are also combinations of
observables for which an uncertainty-principle relationship does not apply,
implying that those observables canbe known simultaneously to any level of
precision. Position and momentum are commonly used to introduce this con-
cept, but the concept is not limited to xand px.
10.5 The Born Interpretation of the
Wavefunction; Probabilities
What we have are two seemingly incompatible ideas. One is that the behavior
of an electron is described by a wavefunction. The other is that the uncertainty
principle limits the certainty with which one can measure various combina-
tions of observables, like position and momentum. How can we discuss the
motion of electrons in any detail at all?
The German scientist Max Born (Figure 10.3) interpreted the wavefunc-
10.5 The Born Interpretation of the Wavefunction; Probabilities 281