Quantum Physics

27.8 The Uncertainty Principle 891waves change in space and time. The Schrödinger wave equation represents a keyelement in the theory of quantum mechanics. It’s as important in quantum mechanicsas Newton’s laws in classical mechanics. Schrödinger’s equation has beensuccessfully applied to the hydrogen atom and to many other microscopic systems.Solving Schrödinger’s equation (beyond the level of this course) determines aquantity called the wave function. Each particle is represented by a wave function that depends both on position and on time. Once is found, 2 gives usinformation on the probability (per unit volume) of finding the particle in anygiven region. To understand this, we return to Young’s experiment involving coherentlight passing through a double slit.First, recall from Chapter 21 that the intensity of a light beam is proportional tothe square of the electric field strength E associated with the beam: I E 2 . Accordingto the wave model of light, there are certain points on the viewing screenwhere the net electric field is zero as a result of destructive interference of wavesfrom the two slits. Because E is zero at these points, the intensity is also zero, andthe screen is dark there. Likewise, at points on the screen at which constructiveinterference occurs, E is large, as is the intensity; hence, these locations are bright.Now consider the same experiment when light is viewed as having a particle nature.The number of photons reaching a point on the screen per second increasesas the intensity (brightness) increases. Consequently, the number of photons thatstrike a unit area on the screen each second is proportional to the square of theelectric field, or N E 2 . From a probabilistic point of view, a photon has a highprobability of striking the screen at a point at which the intensity (and E 2 ) is highand a low probability of striking the screen where the intensity is low.When describing particles rather than photons, rather than E plays the roleof the amplitude. Using an analogy with the description of light, we make the followinginterpretation of for particles: If is a wave function used to describe asingle particle, the value of 2 at some location at a given time is proportional tothe probability per unit volume of finding the particle at that location at that time.Adding up all the values of 2 in a given region gives the probability of finding theparticle in that region.AIP Emilio Segré Visual ArchivesERWIN SCHRÖDINGER, AustrianTheoretical Physicist (1887–1961)Schrödinger is best known as the creator ofwave mechanics, a less cumbersome theorythan the equivalent matrix mechanicsdeveloped by Werner Heisenberg. In 1933Schrödinger left Germany and eventuallysettled at the Dublin Institute of AdvancedStudy, where he spent 17 happy, creativeyears working on problems in general relativity,cosmology, and the application ofquantum physics to biology. In 1956, he returnedhome to Austria and his belovedTirolean mountains, where he died in 1961.27.8 THE UNCERTAINTY PRINCIPLEIf you were to measure the position and speed of a particle at any instant, you wouldalways be faced with experimental uncertainties in your measurements. According toclassical mechanics, no fundamental barrier to an ultimate refinement of the apparatusor experimental procedures exists. In other words, it’s possible, in principle, tomake such measurements with arbitrarily small uncertainty. Quantum theory predicts,however, that such a barrier does exist. In 1927, Werner Heisenberg (1901–1976) introducedthis notion, which is now known as the uncertainty principle:If a measurement of the position of a particle is made with precision x and asimultaneous measurement of linear momentum is made with precision p x ,then the product of the two uncertainties can never be smaller than h/4:x p x [27.16]In other words, it is physically impossible to measure simultaneously the exactposition and exact linear momentum of a particle. If x is very small, then p x islarge, and vice versa.To understand the physical origin of the uncertainty principle, consider the followingthought experiment introduced by Heisenberg. Suppose you wish to measurethe position and linear momentum of an electron as accurately as possible. Youmight be able to do this by viewing the electron with a powerful light microscope.For you to see the electron and determine its location, at least one photon of lightmust bounce off the electron, as shown in Figure 27.18a, and pass through theh4Courtesy of the University of HamburgWERNER HEISENBERG, GermanTheoretical Physicist (1901 – 1976)Heisenberg obtained his Ph.D. in 1923 at theUniversity of Munich, where he studied underArnold Sommerfeld. While physicists such asde Broglie and Schrödinger tried to developphysical models of the atom, Heisenbergdeveloped an abstract mathematical modelcalled matrix mechanics to explain the wavelengthsof spectral lines. Heisenberg mademany other significant contributions tophysics, including his famous uncertaintyprinciple, for which he received the NobelPrize in 1932; the prediction of two forms ofmolecular hydrogen; and theoretical modelsof the nucleus of an atom.