Physical Chemistry , 1st ed.

Heisenberg (Figure 10.2), who announced it in 1927. The uncertainty principle

states that there are ultimate limits to how exact certain measurements can be.

This idea was problematic for many scientists at the time, because science it-

self was concerned with finding specific answers to various questions. Scientists

found that there were limits to how specific those answers could be.

Classically, if you know the position and momentum of a mass at any one

time (that is, if you know those quantities simultaneously), you know everything

about the motion of the mass because you know where it is and where it is go-

ing. If a tiny particle of mass has wave properties and its behavior is described

by a wavefunction, how can we specify its position with a high degree of accu-

racy? According to the de Broglie equation, the de Broglie wavelength is related

to a momentum, but how can we simultaneously determine the position and the

momentum of something with wave behavior? As scientists developed a better

understanding of subatomic matter, it was realized that there are some limits to

the precision with which we can specify two observables simultaneously.

Heisenberg realized this and in 1927 announced his uncertainty principle.

(The principle can be derived mathematically, so it is not a postulate of quan-

tum mechanics. We will not cover the derivation here.) The uncertainty prin-

ciple deals only with certain observables that might be measured simultane-

ously. Two of these observables are position x(in the xdirection), and

momentum px(also in the xdirection). If the uncertainty in the position is

given the symbol xand the uncertainty in the momentum is termed px,

then Heisenberg’s uncertainty principle is

x

px



2

(10.5)

where is h/2. Note the greater-than-or-equal-to sign in the equation. The un-

certainty principle puts a lowerbound on the uncertainty, not an upper bound.

The units of position, m (meters), times the units of momentum, kg m/s, equals

the units on Planck’s constant, J s, which can also be written as kg m^2 /s.

Since the classical definition of momentum pis mv, equation 10.5 is some-

times written as

x m

v



2

(10.6)

where the mass mis assumed to be constant. Equation 10.6 implies that for

large masses, the vand xcan be so small that they are undetectable. However,

for very small masses,xand p(or v) can be so relativelylarge that they

can’t be ignored.

Example 10.5

Determine the uncertainty in position,x, in the following cases:

a.A 1000-kg race car traveling at 100 meters per second, and vis known to

within 1 meter per second.

b.An electron is traveling at 2.00 106 meters per second (the approximate

velocity of an electron in Bohr’s first quantum level) with an uncertainty in

velocity of 1% of the true value.

Solution

a.For an auto traveling at 100 meters per second, an uncertainty of 1 meter

per second is also a 1% uncertainty. The equation for the uncertainty princi-

ple becomes

280 CHAPTER 10 Introduction to Quantum Mechanics

Figure 10.2 Werner Karl Heisenberg (1901–

1976). Heisenberg’s uncertainty principle com-

pletely changed the way science understands the

limitations in the ability to measure nature. In

World War II, Heisenberg was in charge of the

German atomic bomb project, which he appar-

ently purposely delayed to minimize the chance

that the Nazis would develop an atomic bomb.

Max Planck Institut fur Physik, courtesy AIP Emilio Sergre Visual Archives