# Section 6: The Uncertainty Principle

The idea of the position of an object seems so obvious that the concept of position is generally taken for granted in classical physics. Knowing the position of a particle means knowing the values of its coordinates in some coordinate system. The precision of those values, in classical physics, is limited only by our skill in measuring. In quantum mechanics, the concept of position differs fundamentally from this classical meaning. A particle's position is summarized by its wavefunction. To describe a particle at a given position in the language of quantum mechanics, we would need to find a wavefunction that is extremely high near that position and zero elsewhere. The wavefunction would resemble a very tall and very thin tower. None of the wavefunctions we have seen so far look remotely like that. Nevertheless, we can construct a wavefunction that approximates the classical description as precisely as we please.

Let's take the particle in a box described in Section 4 as an example. The possible wavefunctions, each labeled by an integer quantum number, n, obey the superposition principle, and so we are free to add solutions with different values of n, adjusting the amplitudes as needed. The sum of the individual wavefunctions yields another legitimate wavefunction that could describe a particle in a box. If we're clever, we can come up with a combination that resembles the classical solution. If, for example, we add a series of waves with n = 1, 3, 5, and 7 and the carefully chosen amplitudes shown in Figure 22, the result appears to be somewhat localized near the center of the box.

Figure 22: Some wavefunctions for a particle in a box. Curve (e) is the sum of curves (a-d).

Localizing the particle has come at a cost, however, because each wave we add to the wavefunction corresponds to a different momentum. If the lowest possible momentum is p0, then the wavefunction we created has components of momentum at p0, 3p0, 5p0, and 7p0. If we measure the momentum, for instance, by suddenly opening the ends of the box and measuring the time for the particle to reach a detector, we would observe one of the four possible values. If we repeat the measurement many times and plot the results, we would find that the probability for a particular value is proportional to the square of the amplitude of its component in the wavefunction.

If we continue to add waves of ever-shortening wavelengths to our solution, the probability curve becomes narrower while the spread of momentum increases. Thus, as the wavefunction sharpens and our uncertainty about the particle's position decreases, the spread of values observed in successive measurements, that is, the uncertainty in the particle's momentum, increases.

This state of affairs may seem unnatural because energy is not conserved: Often, the particle is observed to move slowly but sometimes it is moving very fast. However, there is no reason energy should be conserved because the system must be freshly prepared before each measurement. The preparation process requires that the particle has the given wavefunction before each measurement. All the information that we have about the state of a particle is in its wavefunction, and this information does not include a precise value for the energy.

The reciprocal relation between the spread in repeated measurements of position and momentum was first recognized by Werner Heisenberg. If we denote the scatter in results for repeated measurements of a position of a particle by (, Greek letter "delta"), and the scatter in results in repeated measurements of the momentum by , then Heisenberg showed that , a result famously known as the Heisenberg uncertainty principle. The uncertainty principle means that in quantum mechanics, we cannot simultaneously know both the position and the momentum of an object arbitrarily well.

Measurements of certain other quantities in quantum mechanics are also governed by uncertainty relations. An important relation for quantum measurements relates the uncertainty in measurements of the energy of a system, , to the time (, Greek letter "tau") during which the measurement is made: .

## Some illustrations of the uncertainty principle

Harmonic oscillator. The ground state energy of the harmonic oscillator, 1/2, makes immediate sense from the uncertainty principle. If the ground state of the oscillator were more highly localized, that is sharper than in Figure 20, the oscillator's average potential energy would be lower. However, sharpening the wavefunction requires introducing shorter wavelength components. These have higher momentum, and thus higher kinetic energy. The result would be an increase in the total energy. The ground state represents the optimum trade-off between decreasing the potential energy and increasing the kinetic energy.

Figure 23: The size of a hydrogen atom is determined by the uncertainty principle.

Hydrogen atom. The size of a hydrogen atom also represents a trade-off between potential and kinetic energy, dictated by the uncertainty principle. If we think of the electron as smeared over a spherical volume, then the smaller the radius, the lower the potential energy due to the electron's interaction with the positive nucleus. However, the smaller the radius, the higher the kinetic energy arising from the electron's confinement. Balancing these trade-offs yields a good estimate of the actual size of the atom. The mean radius is about 0.05 nm.

Natural linewidth. The most precise measurements in physics are frequency measurements, for instance the frequencies of radiation absorbed or radiated in transitions between atomic stationary states. Atomic clocks are based on such measurements. If we designate the energy difference between two states by , then the frequency of the transition is given by Bohr's relation: = . An uncertainty in energy leads to an uncertainty in the transition frequency given by = . The time-energy uncertainty principle can be written , where is the time during which the measurement is made. Combining these, we find that the uncertainty in frequency is .

It is evident that the longer the time for a frequency measurement, the smaller the possible uncertainty. The time may be limited by experimental conditions, but even under ideal conditions would still be limited. The reason is that an atom in an excited state eventually radiates to a lower state by a process called spontaneous emission. This is the process that causes quantum jumps in the Bohr model. Spontaneous emission causes an intrinsic energy uncertainty, or width, to an energy level. This width is called the natural linewidth of the transition. As a result, the energies of all the states of a system, except for the ground states, are intrinsically uncertain. One might think that this uncertainty fundamentally precludes accurate frequency measurement in physics. However, as we shall see, this is not the case.

Figure 24: Gerald Gabrielse (left) is shown with the apparatus he used to make some of the most precise measurements of a single electron.

Myths about the uncertainty principle. Heisenberg's uncertainty principle is among the most widely misunderstood principles of quantum physics. Non-physicists sometimes argue that it reveals a fundamental shortcoming in science and poses a limitation to scientific knowledge. On the contrary, the uncertainty principle is seminal to quantum measurement theory, and quantum measurements have achieved the highest accuracy in all of science. It is important to appreciate that the uncertainty principle does not limit the precision with which a physical property, for instance a transition frequency, can be measured. What it does is to predict the scatter of results of a single measurement. By repeating the measurements, the ultimate precision is limited only by the skill and patience of the experimenter. Should there be any doubt about whether the uncertainty principle limits the power of precision in physics, measurements made with the apparatus shown in Figure 24 should put them to rest. The experiment confirmed the accuracy of a basic quantum mechanical prediction to an accuracy of one part in 1012, one of the most accurate tests of theory in all of science.

## The uncertainty principle and the world about us

Because the quantum world is so far from our normal experience, the uncertainty principle may seem remote from our everyday lives. In one sense, the uncertainty principle really is remote. Consider, for instance, the implications of the uncertainty principle for a baseball. Conceivably, the baseball could fly off unpredictably due to its intrinsically uncertain momentum. The more precisely we can locate the baseball in space, the larger is its intrinsic momentum. So, let's consider a pitcher who is so sensitive that he can tell if the baseball is out of position by, for instance, the thickness of a human hair, typically 0.1 mm or 10-4 m. According to the uncertainty principle, the baseball's intrinsic speed due to quantum effects is about 10-29 m/s. This is unbelievably slow. For instance, the time for the baseball to move quantum mechanically merely by the diameter of an atom would be roughly 20 times the age of the universe. Obviously, whatever might give a pitcher a bad day, it will not be the uncertainty principle.

Figure 25: The effect of quantum mechanical jitter on a pitcher, luckily, is too small to be observable.