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12.6
Probability versus Determinism
Heisenberg’s Uncertainty Principle
Figure 1
Werner Heisenberg (1901–1976) was
awarded the 1932 Nobel Prize in
physics for developing the uncertainty principle.
Once scientists had abandoned the well-established realm of classical Newtonian
mechanics and entered the new, uncharted world of quantum mechanics, many phenomena were discovered that were both strange and difficult to visualize. In classical
mechanics, objects we identify as particles always behave like particles, and wave phenomena always exhibit pure wave properties. But the quantum hypotheses of Planck,
Einstein, and Bohr created a new dilemma: light, which had traditionally been viewed
as a wave phenomenon, was apparently composed of photons possessing distinct particle characteristics. Electrons, to this point thought of as tiny particles with a definite
charge and mass, behaved like waves with a definite wavelength, as they moved in orbits
within the atom and interacted with objects of atomic dimensions. Clearly, a new way
of looking at particle and wave phenomena was needed.
The equations in Section 12.5 suggest that we can be very precise about the properties of an electron in the hydrogen atom. We can determine an orbiting electron’s exact
location using the equation for rn and can likewise determine its exact speed and energy.
In 1927, the German physicist Werner Heisenberg (Figure 1) proposed that there was
always some inherent uncertainty in the determination of these quantities. It was not a
question of accuracy in measurements. The uncertainty was inherent and originated
from the quantum-mechanical nature of subatomic particles. Let us consider an illustration of Heisenberg’s position.
Suppose we are measuring the temperature of a cup of hot coffee with a laboratory
thermometer. When the room-temperature thermometer is immersed in the coffee, it
extracts some heat from the coffee. Therefore, the temperature registered is that of the
system and not just the original temperature of the coffee. A smaller thermometer will
disturb the temperature of the coffee less, thereby yielding a more precise reading. In the
limiting situation, an infinitely small thermometer would produce an infinitely precise
result. There is no inherent lower limit to the size of the thermometer that would limit
the precision of the temperature determination.
Imagine, next, that we are trying to determine the position of an electron within
an atom. To do this, we bombard the atom with highly energetic electromagnetic
radiation of a given wavelength (Figure 2). A photon of wavelength λ will have a
short λ photon
long λ photon
scattered photon
scattered photon
electron
electron
∆x 2
∆x1
Figure 2
The more accurately we try to locate
the electron, the less precisely we
know its momentum.
650 Chapter 12
p + ∆p 2
p + ∆p 1
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Section 12.6
h
momentum . We know from our analysis of the Compton effect (Section 12.3) that
λ
this photon will transfer some of its momentum to the electron when they interact.
Thus, the electron will acquire a new and unknown value of momentum just as a consequence of being located by the photon. The more accurately we try to locate the electron by selecting a photon of smaller wavelength, the less precisely we know its
momentum. If we use a photon of longer wavelength to be more precise about the
momentum of the electron, we must be content with being less certain of its location.
As a result, we are unable to measure both the position and the momentum of the
electron with unlimited accuracy. Heisenberg was able to determine the limits of these
inherent uncertainties and to express them mathematically, in a formulation known as
Heisenberg’s uncertainty principle:
Heisenberg’s Uncertainty Principle
If ∆x is the uncertainty in a particle’s position, and ∆p is the
uncertainty in its momentum, then
h
xp where h is Planck’s constant
2p
To appreciate the significance of Heisenberg’s principle, let us look at examples from
both the macroscopic and the microscopic worlds. For a large particle of mass 1.0 kg,
whose momentum p is given by the equation p mv,
h
∆x∆p ∆x m ∆v 2p
h
∆x∆v 2pm
6.63 1034 Js
2p(1.0 kg)
∆x∆v 1.1 1034 Js/kg
Thus, if we can know the position of the particle to within an uncertainty ∆ x of
106 m, we are permitted to know its speed to within an uncertainty ∆v
of 1028 m/s (since the product of ∆x∆v cannot be smaller than 1034 Js/kg). Such
small uncertainties are of no consequence when dealing with macroscopic objects.
On the other hand, for an electron of mass 9.1 1031 kg,
h
∆x∆v 2pm
6.63 1034 Js
2p(9.1 1031 kg)
∆x∆v 1.2 104 Js/kg
If we confine the uncertainty in the location of the electron to 1010 m (about the
size of an atom), we cannot know its speed within an uncertainty of less than 106 m/s.
These uncertainties are significant enough to invalidate Bohr’s picture of a finite electron moving in a well-defined orbit. Clearly, a new look is needed at just what an electron is. We will leave this discussion to Chapter 13.
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Waves, Photons, and Matter
651
Probability and Determinism
LEARNING
TIP
Determinism
Determinism is the philosophical doctrine that every event,
act, or decision is the inevitable
consequence of antecedents
that are independent of the
human will.
Nelson Canadian Dictionary
The Newtonian view of mechanics was deterministic. If the position and speed of an
object at a particular time were known, its position and speed for all future time could
be determined simply by knowing the forces acting on it. More than two centuries of success in describing the macroscopic world had made the deterministic view of nature
convincing. Bohr had no reason to believe that the motion of electrons within atoms
would be any different.
However, because of the uncertainty in measuring an electron’s position and speed,
it becomes impossible to say, for any individual electron, where it is now or where it will
be at any future time. Because of its inherent wave properties, an electron in an atom
cannot be visualized as a Newtonian particle. Rather than saying where it is, we must be
satisfied with describing its location by stating the probability that it will be found near
any point.
One way of visualizing this situation is to think of the electron existing as a cloud of
negative charge distributed around the atom rather than as a particle moving in a circular orbit (Figure 3). The cloud is denser in areas of high probability and less dense in
areas where the electron is less likely to be found. For the first Bohr orbit of the hydrogen
atom (n 1), the cloud is most dense in a spherical shell of radius 5.3 1011 m, the
first Bohr radius. Even though this is where the electron is most likely to be, it does have
a small statistical probability of being anywhere else.
high probability
of finding electron
Figure 3
The electron cloud, or probability
distribution, for the ground state of
the hydrogen atom. The cloud is
most dense—corresponding to the
highest probability—at a distance
from the nucleus of 5.3 1011 m,
which is what the Bohr model predicts for the radius of the first orbit.
But modern quantum mechanics
departs from the Bohr model in
telling us that the electron can be
within or beyond that distance at
any given time.
nucleus
low probability
of finding electron
The “probability cloud” for other allowed Bohr orbits takes on different shapes and
dimensions in ways that can be determined mathematically. In 1925, working independently, Heisenberg and Erwin Schrödinger devised equivalent mathematical equations whose solutions gave the probability distributions for virtually any problem in
quantum mechanics. The expressions became known as the Schrödinger equations and
have been immensely successful in solving atomic structure problems. They provide a
mathematical interpretation of the physical situation, yielding the values of all measurable quantities such as momentum and energy, as well as the probability for all positions that the particle can occupy. Physicists have spent countless hours applying these
techniques to a seemingly endless array of problems, with success in almost every instance.
One example is the whole field of solid-state physics, from which semiconductors, the
backbone of modern electronics, have developed.
652 Chapter 12
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Section 12.6
SUMMARY
•
Probability versus Determinism
In classical mechanics, the objects we identify as particles always behave like particles, while wave phenomena always exhibit pure wave properties.
•
In quantum mechanics, light is composed of photons possessing distinct particle
characteristics, and electrons behave like waves with a definite wavelength.
•
Heisenberg proposed that an inherent uncertainty exists in the simultaneous
determination of any measured quantity due to the quantum-mechanical wave
aspect of particles.
•
We are unable to measure both the position and the momentum of the electron
with unlimited accuracy. Heisenberg was able to determine the limits of these
h
inherent uncertainties and to express them mathematically as ∆x ∆p .
2p
Physics now considers the electron in an atom as a probability cloud of negative
charge distributed around the nucleus rather than as a particle moving in a circular orbit. The cloud is denser in areas of high probability and less dense in
areas where the electron is less likely to be found.
•
•
DID YOU
KNOW
?
Uncertainty
Einstein, who had trouble
accepting the uncertainty principle, was quoted as saying: “God
does not play dice with the
Universe.” The contemporary cosmologist Steven Hawking, on the
other hand, has said: “Not only
does God play dice with the
Universe, he sometimes throws
them where they can’t be seen.”
(in A Brief History of Time, with reference to black holes)
An electron’s position and velocity are impossible to predict; we only know the
probability that the electron will be found near any point. The shapes and dimensions of these probability distributions can be determined mathematically.
Section 12.6 Questions
Understanding Concepts
1. In what sense does Bohr’s model of the atom violate the
uncertainty principle?
2. How accurately can the position and speed of a particle be
known, simultaneously?
3. A 12.0-g bullet leaves a rifle with a speed of 1.80 102 m/s.
(a) Calculate the de Broglie wavelength associated with
the bullet.
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(b) If the position of the bullet is known to an accuracy of
0.60 cm (radius of the rifle barrel), what is the minimum uncertainty in momentum?
4. If Planck’s constant were very large, what would be the
effect on shooting an arrow at a stationary target?
What would be the effect with a moving target, such
as a deer?
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