BioPHYSICAL chemistry

A quantum-mechanical world

Under common circumstances, people do not encounter effects such as
the wavelength nature of matter or the Uncertainty Principle, due to
the size and speed of everyday objects. To better understand the issues
raised by these concepts, consider a quantum-mechanical world where
everyone is so small that the wave nature of matter must be dealt with
in common activities. The wave nature of matter becomes important when
h/pyields a wavelength that is the size of the objects. To achieve this
limit, let’s have a population of people whose height is about the size of
an electron orbital, or 0.1 nm, and who have a mass of me. If the people
can run at a speed of 7 × 106 ms−^1 , then the wavelength is the same as
the height and the effects are observed daily. For example, when a person
walks through a door that has an opening of 0.1 nm, then the person
will not be able to walk through the door in a straight line but will undergo
diffraction.

(9.57)

= 10 −^10 m=0.1 nm

To illustrate the complexities, consider people in this quantum-
mechanical world playing the sport baseball. A ball is hit deep
and the outfielder chases after the ball. He gets to the spot
where the trajectory predicts that the ball should land but
can he catch the ball? To catch the ball he must be able to
predict the landing position of the ball within the accuracy
determined by the size of his glove, which has a size of 0.01 nm.
Although the overall trajectory can be predicted, the path
becomes fuzzy due to the wave nature of the ball, and the pre-
dicted landing spot is limited by the Heisenberg Uncertainty
Principle. If the ball has a mass of 0.1 meand the speed has
an uncertainty of 5 × 105 ms−^1 , then the predicted landing
spot will have an uncertainty of 0.1 nm. Since the uncertainty
in the landing spot is 10 times larger than the size of the
glove, then the outfielder will have only a 10% chance of
catching the ball. This example, of course, does not address the
other difficulties, including hitting the ball in the first place
(Figure 9.9).

= 10 −^10 m=0.1 nm (9.58)

Δ

Δ

x

()px

.

(.

==

×

×

−

−

Z

2

11 10

2 9 109 10

34

31

Js

kkg ×^51 m s )(− ) 510

λ

.

(. )(

==

×

×

−

−

h

mv

66 10

9 109 10 7

34

31

Js

kg ×× 1061 ms−)

CHAPTER 9 QUANTUM THEORY 193

Figure 9.9Outfielders have

a difficult time catching a

baseball in a quantum-

mechanical world.