102 Probability: The Heisenberg Uncertainty Principle

Probability Distribution

Matter and photons display wave properties, which means that their behavior is spread over a region of space rather than being confined to a single classical path. This raises an important question: what does it mean to describe the position of a particle such as an electron? Is the electron located at the center of its associated wave, or is it distributed throughout the wave? Experiments show that whenever an electron is detected, it is found at one definite location. A detector does not record part of an electron spread across several places. Instead, it records a single localized event, such as one point on a screen. In this sense, each individual detection is particle-like. However, suppose an experiment is prepared in exactly the same way and repeated many times. The electron will not necessarily be detected at the same location in every trial. It may appear at one location during the first trial, at another location during the next trial, and somewhere else during a later trial. These differences are not simply the result of poor instruments or ordinary experimental error. Even when the preparation and measurement are controlled as carefully as possible, quantum mechanics does not generally predict the exact location of an individual detection. Although the position of any one electron cannot be predicted precisely, the results of many repeated measurements form a regular and predictable statistical pattern. Some locations receive many electrons, while other locations receive few or none. The resulting distribution has the same form as the diffraction or interference pattern associated with a wave, as shown in Figure 102.1.

The distinction between an individual detection and the complete pattern is essential. Quantum mechanics predicts the probability that a particle will be detected within a particular region. It does not ordinarily predict the exact result of one individual measurement. A region in which many electrons are detected corresponds to a high probability, while a region in which few electrons are detected corresponds to a low probability. The overall spatial pattern describing these probabilities is called a probability distribution. A probability distribution does not mean that a fraction of the electron exists at every possible position. Rather, it describes the relative likelihood of obtaining each possible position when the same measurement is performed repeatedly on identically prepared particles. After de Broglie proposed that matter has wave properties, physicists including Erwin Schrödinger and Werner Heisenberg explored the consequences of this idea. One of the most important conclusions was that, because of its wave character, the precise trajectory and final destination of an individual particle generally cannot be predicted. Nevertheless, each measurement produces one definite result. After enough measurements have been collected, the results form a probability distribution related to the particle’s wavelength and its corresponding diffraction pattern.

The relationship between wave behavior and probability becomes especially clear in a double-slit experiment. Figure 102.2 compares the interference patterns produced by two different types of particles. When the particles have equal wavelengths and pass through slits with the same separation, they produce identical interference patterns.

The locations of the bright interference regions follow the double-slit constructive-interference condition

where [latex]d[/latex] is the separation between the slits, [latex]\theta[/latex] is the angle at which a bright region appears, [latex]m[/latex] is an integer, and [latex]\lambda[/latex] is the wavelength of the electron, proton, photon, or other particle. The interference pattern develops statistically as individual particles strike the detector. At first, the detected particles appear as isolated points with no obvious pattern. As more particles arrive, alternating regions of high and low detection probability gradually emerge. After a sufficiently large number of particles have been recorded, the familiar interference pattern becomes clear. It might initially seem that electrons produce this pattern by interfering with other electrons in the beam. This possibility can be tested by reducing the beam intensity until only one electron at a time is present between the source and the detector. Under these conditions, no electron can interact with another electron while passing through the apparatus. Nevertheless, the same interference pattern gradually develops. This result shows that the pattern is not caused by collisions or interactions between different electrons. Instead, the quantum state associated with each electron has wave-like properties and extends through the experimental arrangement. The possible alternatives represented by the two slits combine according to the rules of wave interference. This behavior is often described by saying that the electron’s probability distribution interferes with itself. This does not mean that an electron divides into two smaller pieces. An electron is detected as one complete particle at one location. The wave-like aspect describes the possible outcomes and their probabilities, while the detector records a localized particle-like event. We might try to determine which slit the electron passes through by placing detectors near the openings. When a measurement is made that identifies the path, each electron is found to pass through one slit or the other. However, there is an important consequence: once which-path information is obtained, the double-slit interference pattern disappears. The resulting distribution is instead consistent with the patterns produced by the two slits acting independently. This outcome does not depend on the particular device used to determine the path. Any measurement that makes it possible to distinguish which slit the particle used changes the experimental situation. When the two paths remain indistinguishable, their probability amplitudes interfere. When the paths become distinguishable through measurement, the interference between them is lost.

Heisenberg Uncertainty

Why does obtaining information about an electron’s path change the observed pattern? The answer reflects a central feature of quantum mechanics: measurement is a physical interaction with the system being measured. To determine the position of a moving electron, something must interact with it. For example, a photon may scatter from the electron and then travel to a detector, or another particle may collide with it. The measuring probe carries momentum and energy. During the interaction, some momentum or energy is exchanged with the electron, altering its state. This effect is especially important for microscopic particles because the momentum carried by the measuring probe may be comparable to the momentum of the particle being observed. The measurement therefore cannot be treated as an entirely passive act. Information about one property may be obtained only at the cost of losing precision in another property. This limitation is not merely a practical problem caused by imperfect instruments. Quantum mechanics places a fundamental limit on the simultaneous precision with which certain pairs of physical quantities can be specified. Position and momentum are one such pair.

Werner Heisenberg formulated this limit in 1927 while developing quantum mechanics and studying the wave properties of particles. Consider the simultaneous measurement of the position and momentum of an electron. The same reasoning applies to any quantum particle. Because a wave extends over a region of space rather than occupying one geometric point, a particle described by a wavelength [latex]\lambda[/latex] cannot be localized much more precisely than approximately one wavelength. We may therefore estimate the uncertainty in position as

Here, [latex]\Delta x[/latex] represents the spread in possible position measurements. If the position measurement is repeated using identically prepared particles, the measured locations will occupy a range rather than producing one identical result every time. To detect the particle, it must interact with a measuring device. During this interaction, momentum is exchanged. The amount of momentum transferred may vary from nearly zero to an amount comparable to the particle’s original momentum. Because the exact momentum transfer cannot be known simultaneously with a precise position measurement, the particle also has an uncertainty in momentum, represented by [latex]\Delta p[/latex]. Using the de Broglie relationship

we can estimate the momentum uncertainty as

These two approximate relationships reveal an unavoidable trade-off. A shorter wavelength allows the particle to be localized within a smaller region, reducing [latex]\Delta x[/latex]. However, the shorter wavelength corresponds to a larger momentum and therefore to a larger uncertainty in momentum. Conversely, using a longer wavelength reduces the momentum uncertainty but spreads the wave over a larger region of space, increasing the uncertainty in position. Multiplying the two approximate uncertainty relationships gives

The wavelength cancels, leaving

This approximate result shows that reducing one uncertainty necessarily increases the other. More advanced quantum-mechanical analysis gives the precise lower limit

This relationship is known as the Heisenberg uncertainty principle. It states that position and momentum cannot simultaneously be known with uncertainties whose product is smaller than [latex]h/(4\pi)[/latex]. Neither uncertainty can be zero. A particle cannot possess both an exactly defined position and an exactly defined momentum at the same time. If the position is specified with greater precision, the range of possible momentum values must increase. If the momentum is specified with greater precision, the range of possible positions must increase.

A short-wavelength probe can provide precise position information because it can resolve a small region of space. However, a short wavelength corresponds to a large momentum. When such a probe interacts with a particle, the momentum exchange can substantially alter the particle’s motion. For example, suppose an electron is used to investigate another electron within an atom. To resolve distances comparable to the size of the atom, the probing electron must have a very short wavelength and therefore substantial momentum. Its interaction with the atomic electron may transfer enough energy and momentum to remove that electron from the atom. The measurement may reveal a position, but it destroys the original state whose motion we hoped to observe. For this reason, it is not physically meaningful to imagine following an electron as it travels around the nucleus along a precise classical orbit. An electron in an atom does not possess both a sharply defined trajectory and a sharply defined position at every instant. If the position of an electron is measured, one definite location is obtained. However, repeated measurements performed on many identically prepared atoms produce a spatial probability distribution. These distributions are called electron clouds or atomic orbitals. An orbital should not be interpreted as a path traveled by an electron. Instead, it represents the probability of detecting the electron in different regions surrounding the nucleus. Regions of greater probability are often drawn more densely or enclosed by surfaces that contain a chosen percentage of the total probability. These orbital probability distributions are central to chemistry and the health sciences. Their shapes and energies help explain chemical bonding, molecular geometry, electron transfer, and the interactions that determine the structures and functions of biological molecules.

Why do we not notice the Heisenberg uncertainty principle in everyday life? The reason is that Planck’s constant is extremely small:

Because the lower bound in the uncertainty relation contains this very small constant, quantum uncertainty is most significant for microscopic particles with small masses and momenta. For macroscopic objects, the minimum uncertainty is usually far too small to detect. For example, sunlight reflected from Jupiter allows astronomers to determine the planet’s position and follow its orbit. The photons that strike and leave Jupiter transfer some momentum to the planet, but Jupiter’s mass and momentum are so enormous that the resulting change and uncertainty are negligible. The same is true for ordinary biological and medical objects. A cell, organ, patient, surgical instrument, or imaging detector contains an immense number of particles and has a mass that is enormous compared with the mass of an electron. Its quantum uncertainty in position or momentum is therefore negligible on the scale of ordinary measurements. This agreement between quantum mechanics and classical physics for large systems is an example of the correspondence principle. Quantum mechanics remains valid, but its predictions become effectively indistinguishable from classical physics when applied to sufficiently large objects.

Heisenberg Uncertainty for Energy and Time

There is another important form of Heisenberg’s uncertainty principle that relates energy and time:

Here, [latex]\Delta E[/latex] is the uncertainty or spread in energy, and [latex]\Delta t[/latex] is a characteristic time interval. Depending on the situation, [latex]\Delta t[/latex] may represent the duration of a measurement or the lifetime of a particular quantum state. The relationship means that determining an energy very precisely requires observing the system for a sufficiently long time. A short observation time is associated with a larger possible spread in the measured energy. Conversely, a state that persists for a long time can have a more sharply defined energy. This form of the uncertainty principle is especially useful for unstable or short-lived states. An atom in an excited state, for example, remains in that state only briefly before emitting a photon and moving to a lower-energy state. Because the excited state has a finite lifetime, its energy cannot be perfectly sharp. Instead, repeated measurements reveal a small range of energies. The energy-time uncertainty relation is often written using the reduced Planck constant, [latex]\hbar=h/(2\pi)[/latex], as

This is mathematically equivalent to the form written with [latex]h[/latex>.

The energy-time uncertainty principle becomes especially important for systems that exist only briefly. If the lifetime of a quantum state is very short, then [latex]\Delta t[/latex] is small, and the corresponding uncertainty in energy [latex]\Delta E[/latex] is relatively large. Some unstable nuclei and elementary particles have lifetimes as short as [latex]10^{-25}\,\text{s}[/latex], producing energy uncertainties of several GeV. Because mass and energy are related through Einstein's equation, [latex]E=mc^2[/latex], a spread in energy also corresponds to a spread in the measured mass of a short-lived particle. When many identical particles are produced and measured, they do not all appear to have exactly the same mass. Instead, the measured masses form a narrow distribution whose width is directly related to the particle's lifetime. This relationship is extremely useful in experimental particle and nuclear physics. Some unstable particles decay so rapidly that their lifetimes cannot be measured directly. Instead, physicists measure the spread in the observed decay energies or masses and use the energy-time uncertainty principle to determine the particle's lifetime.

Throughout this chapter we have seen that individual quantum measurements always produce definite outcomes. An electron, for example, is detected at one specific location each time it is measured. However, when many identical experiments are repeated, those individual results build up a probability distribution whose shape reflects the wave nature of matter. The physicist Richard Feynman summarized this idea by remarking, "What there are, are particles." Individual observations always detect particles, yet the collection of many observations reveals the statistical wave behavior predicted by quantum mechanics. Exactly how a quantum system evolves between measurements cannot be described using the same classical concepts of trajectories that work for everyday objects.