29.6

The Heisenberg Uncertainty Principle

As the previous section discusses, the bright fringes in Figure 29-14 indicate the places where there is a high probability of an electron striking the screen. And since there are a number of bright fringes, there is more than one place where each electron has some probability of hitting. As a result, it is not possible to specify in advance exactly where on the screen an individual electron will hit. All we can do is speak of the probability that the electron may end up in a number of different places. No longer is it possible to say, as Newton’s laws would suggest, that a single electron, fired through the double slit, will travel directly forward in a straight line and strike the screen. This simple model just does not apply when a particle as small as an electron passes through a pair of closely spaced narrow slits. Because the wave nature of particles is important in such circumstances, we lose the ability to predict with 100% certainty the path that a single particle will follow. Instead, only the average behavior of large numbers of particles is predictable, and the behavior of any individual particle is uncertain.
To see more clearly into the nature of the uncertainty, consider electrons passing through a single slit, as in Figure 29-15. After a sufficient number of electrons strike the screen, a diffraction pattern emerges. The electron diffraction pattern consists of alternating bright and dark fringes and is analogous to that for light waves shown in Figure 27-24. Figure 29-15 shows the slit and locates the first dark fringe on either side of the central bright fringe. The central fringe is bright because electrons strike the screen over the entire region between the dark fringes. If the electrons striking the screen outside the central bright fringe are neglected, the extent to which the electrons are diffracted is given by the angle in the drawing. To reach locations within the central bright fringe, some electrons must have acquired momentum in the y direction, despite the fact that they enter the slit traveling along the x direction and have no momentum in the y direction to start with. The figure illustrates that the y component of the momentum may be as large as . The notation indicates the difference between the maximum value of the y component of the momentum after the electron passes through the slit and its value of zero before the electron passes through the slit. represents the uncertainty in the y component of the momentum in that the y component may have any value from zero to .

Figure 29-15 When a sufficient number of electrons pass through a single slit and strike the screen, a diffraction pattern of bright and dark fringes emerges. (Only the central bright fringe is shown.) This pattern is due to the wave nature of the electrons and is analogous to that produced by light waves.

It is possible to relate to the width W of the slit. To do this, we assume that Equation 27.4, which applies to light waves, also applies to particle waves whose de Broglie wavelength is . This equation, , specifies the angle that locates the first dark fringe. If is small, then . Moreover, Figure 29-15 indicates that , where px is the x component of the momentum of the electron. Therefore, . But according to de Broglie’s equation, so that

As a result,

| (29.9)

which indicates that a smaller slit width leads to a larger uncertainty in the y component of the electron’s momentum.
It was Heisenberg who first suggested that the uncertainty in the y component of the momentum is related to the uncertainty in the y position of the electron as the electron passes through the slit. Since the electron can pass through anywhere within the width W, the uncertainty in the y position of the electron is . Substituting for W in Equation 29.9 shows that or . The result of Heisenberg’s more complete analysis is given below in Equation 29.10 and is known as the Heisenberg uncertainty principle. Note that the Heisenberg principle is a general principle with wide applicability. It does not just apply to the case of single-slit diffraction, which we have used here for the sake of convenience.

| The Heisenberg Uncertainty Principle
| Momentum and position

| (29.10)

= uncertainty in a particle’s position along the y direction
= uncertainty in the y component of the linear momentum of the particle
Energy and time

| (29.11)

= uncertainty in the energy of a particle when the particle is in a certain state
= time interval during which the particle is in the state

The Heisenberg uncertainty principle places limits on the accuracy with which the momentum and position of a particle can be specified simultaneously. These limits are not just limits due to faulty measuring techniques. They are fundamental limits imposed by nature, and there is no way to circumvent them. Equation 29.10 indicates that and cannot both be arbitrarily small at the same time. If one is small, then the other must be large, so that their product equals or exceeds () Planck’s constant divided by . For example, if the position of a particle is known exactly, so that is zero, then is an infinitely large number, and the momentum of the particle is completely uncertain. Conversely, if we assume that is zero, then is an infinitely large number, and the position of the particle is completely uncertain. In other words, the Heisenberg uncertainty principle states that it is impossible to specify precisely both the momentum and position of a particle at the same time.
There is also an uncertainty principle that deals with energy and time, as expressed by Equation 29.11. The product of the uncertainty in the energy of a particle and the time interval during which the particle remains in a given energy state is greater than or equal to Planck’s constant divided by . Therefore, the shorter the lifetime of a particle in a given state, the greater is the uncertainty in the energy of that state.
Example 6 shows that the uncertainty principle has significant consequences for the motion of tiny particles such as electrons but has little effect on the motion of macroscopic objects, even those with as little mass as a Ping-Pong ball.

| Example 6 | | | The Heisenberg Uncertainty Principle

Assume that the position of an object is known so precisely that the uncertainty in the position is only . (a) Determine the minimum uncertainty in the momentum of the object. Find the corresponding minimum uncertainty in the speed of the object in the case when the object is (b) an electron () and (c) a Ping-Pong ball ().
Reasoning The minimum uncertainty in the y component of the momentum is given by the Heisenberg uncertainty principle as , where is the uncertainty in the position of the object along the y direction. Both the electron and the Ping-Pong ball have the same uncertainty in their momenta because they have the same uncertainty in their positions. However, these objects have very different masses. As a result, we will find that the uncertainty in the speeds of these objects is very different.

Problem solving insight The Heisenberg uncertainty principle states that the product of and is greater than or equal to . The minimum uncertainty occurs when the product is equal to .

Solution

a. | The minimum uncertainty in the y component of the momentum is

| (29.10)

b. | Since , the minimum uncertainty in the speed of the electron is

Thus, the small uncertainty in the y position of the electron gives rise to a large uncertainty in the speed of the electron.
c. | The uncertainty in the speed of the Ping-Pong ball is

Because the mass of the Ping-Pong ball is relatively large, the small uncertainty in its y position gives rise to an uncertainty in its speed that is much smaller than that for the electron. Thus, in contrast to the electron, we can know simultaneously where the ball is and how fast it is moving, to a very high degree of certainty.

Example 6 emphasizes how the uncertainty principle imposes different uncertainties on the speeds of an electron (small mass) and a Ping-Pong ball (large mass). For objects like the ball, which have relatively large masses, the uncertainties in position and speed are so small that they have no effect on our ability to determine simultaneously where such objects are and how fast they are moving. The uncertainties calculated in Example 6 depend on more than just the mass, however. They also depend on Planck’s constant, which is a very small number. It is interesting to speculate about what life would be like if Planck’s constant were much larger than . Conceptual Example 7 deals with just such speculation.

| Conceptual Example 7 | | | What If Planck’s Constant Were Large?

A bullet leaving the barrel of a gun is analogous to an electron passing through the single slit in Figure 29-15. With this analogy in mind, what would hunting be like if Planck’s constant had a relatively large value instead of its small value of ?
Reasoning and Solution It takes two to hunt, the hunter and the hunted. Consider the hunter first and what it would mean to take aim if Planck’s constant were large. The bullet moving down the barrel of the gun has a de Broglie wavelength given by Equation 29.8 as . This wavelength is large, since Planck’s constant h is now large. (We are assuming that the magnitude p of the bullet’s momentum is not abnormally large.) Remember from Section 27.5 that the diffraction of light waves through an opening is greater when the wavelength is greater, other things being equal. We expect, therefore, that with being comparable to the opening in the barrel, the hunter will have to contend with appreciable diffraction when the bullet leaves the barrel. This means that, in spite of being aimed directly at the target, the bullet may come nowhere near it. In Figure 29-15, for comparison, diffraction allows the electron to strike the screen on either side of the maximum intensity point of the central bright fringe. Under such circumstances, the hunter might as well not bother aiming. He’ll have just as much success by scattering his shots around randomly, hoping to hit something.
Now consider the target and what it means that the hunter “aimed directly at the target.” This must mean that the target’s location is known, at least approximately. In other words, the uncertainty in the target’s position is reasonably small. But the uncertainty principle indicates that the minimum uncertainty in the target’s momentum is . Since Planck’s constant is now large, is also large. The target, therefore, may have a large momentum and be moving rapidly. Even if the bullet happens to reach the place where the hunter aimed, the target may no longer be there. Hunting would be vastly different indeed, if Planck’s constant had a very large value.
Related Homework: Problem 34

Test your understanding of the material in Sections 29.4, 29.5 and 29.6 :

· | The Momentum of a Photon
· | The Compton Effect
· | The De Broglie Wavelength and the Wave Nature of Matter
· | The Heisenberg Uncertainty Principle

Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.