Chapter 1 The Schrodinger Equation 1.1 Quantum Chemistry In the late seventeenth century.Isaac Newton discovered classical mechanics,the laws of motion of macroscopic objects.In the early twentieth century,physicists found that classi- cal mechanics does not correctly describe the behavior of very small particles such as the electrons and nuclei of atoms and molecules.The behavior of such particles is described by a set of laws called quantum mechanics. Quantum chemistry applies quantum mechanics to problems in chemistry.The influence of quantum chemistry is evident in all branches of chemistry.Physical chem- ists use quantum mechanics to calculate (with the aid of statistical mechanics)thermo dynamic properties (or example cap to interpret molecular nation of e mo of rate c stand inter lar fo and to deal with b Organic chemists use au echan ies to stimate the relati stabilities of mol- ecules to calculate n operties of reaction intermediates,to investigate the mechanisms of chemical reactions and to analyze and predict nuclear-magnetic-resonance spectra Analytical chemists use spectroscopic methods extensively.The frequencies and in- tensities of lines in a spectrum can be properly understood and interpreted only through the use of quantum mechanics. Inorganic chemists use ligand field theory,an approximate quantum-mechanical method,to predict and explain the properties of transition-metal complex ions. Although the large size of biologically important molecules makes quantum mechanical calculations on them extremely hard.biochemists are beginning to benefit e stud g,and tion bi0fconorm tions of biological molecules,enzyme- om aterials (objec with at leas ne e be ed.Whe more di ons of rial fall b P al and other operties from those of the bulk material ca metal obiect with one dimension in the I to 100 nm range is calleda well,one with two dimensions in this range is a quantum wire:and one with all three dimensions in this range is a quantum dot.The word quantum in these names indicates the key role played by quantum mechanics in determining the properties of such materials.Many
1 Chapter 1 The Schrödinger Equation 1.1 Quantum Chemistry In the late seventeenth century, Isaac Newton discovered classical mechanics, the laws of motion of macroscopic objects. In the early twentieth century, physicists found that classical mechanics does not correctly describe the behavior of very small particles such as the electrons and nuclei of atoms and molecules. The behavior of such particles is described by a set of laws called quantum mechanics. Quantum chemistry applies quantum mechanics to problems in chemistry. The influence of quantum chemistry is evident in all branches of chemistry. Physical chemists use quantum mechanics to calculate (with the aid of statistical mechanics) thermodynamic properties (for example, entropy, heat capacity) of gases; to interpret molecular spectra, thereby allowing experimental determination of molecular properties (for example, molecular geometries, dipole moments, barriers to internal rotation, energy differences between conformational isomers); to calculate molecular properties theoretically; to calculate properties of transition states in chemical reactions, thereby allowing estimation of rate constants; to understand intermolecular forces; and to deal with bonding in solids. Organic chemists use quantum mechanics to estimate the relative stabilities of molecules, to calculate properties of reaction intermediates, to investigate the mechanisms of chemical reactions, and to analyze and predict nuclear-magnetic-resonance spectra. Analytical chemists use spectroscopic methods extensively. The frequencies and intensities of lines in a spectrum can be properly understood and interpreted only through the use of quantum mechanics. Inorganic chemists use ligand field theory, an approximate quantum-mechanical method, to predict and explain the properties of transition-metal complex ions. Although the large size of biologically important molecules makes quantummechanical calculations on them extremely hard, biochemists are beginning to benefit from quantum-mechanical studies of conformations of biological molecules, enzyme– substrate binding, and solvation of biological molecules. Quantum mechanics determines the properties of nanomaterials (objects with at least one dimension in the range 1 to 100 nm), and calculational methods to deal with nanomaterials are being developed. When one or more dimensions of a material fall below 100 nm (and especially below 20 nm), dramatic changes in the optical, electronic, chemical, and other properties from those of the bulk material can occur. A semiconductor or metal object with one dimension in the 1 to 100 nm range is called a quantum well; one with two dimensions in this range is a quantum wire; and one with all three dimensions in this range is a quantum dot. The word quantum in these names indicates the key role played by quantum mechanics in determining the properties of such materials. Many
2 Chapter 1 The Schrodinger Equation people have speculated that nanoscience and nanotechnology will bring about the"next industrial revolution." The rapid increase in computer speed and the development of new methods(such as density functional theory-Section 16.4)of doing molecular calculations have made quantum chemistry a practical tool in all areas of chemistry.Nowadays,several compa- nies sell quantum-chemistry software for doing molecular quantum-chemistry calcula- tions.These programs are designed to be used by all kinds of chemists,not just quantum elan Because of the rapi ly expanding role of qua mical Society b stry and I.the American CH ournal of Che on,in s nea nc 1.2 Historical Background of Quantum Mechanics The development of quantum mechanics began in 1900 with Planck's study of the light idence for th e nature of ligh b erving dif Thomas Young gave convincing erterence whe t through two (Diffraction s the be ng of a wave aroun an ot Interference is the s the ale of the h spac n e.See any first- hysies text In 1864.James Clerk Maxwell published four equations,known as Maxwell's equa tions which unified the laws of electricity and magnetism maxwell's equations predicted that an accelerated electric charge would radiate energy in the form of electromagnetic waves consisting of oscillating electric and magnetic fields.The speed predicted by Max- well's equations for these waves turned out to be the same as the experimentally measured speed of light.Maxwell concluded that light is an electromagnetic wave. In 1888.Heinrich Hertz detected radio waves produced by accelerated electric charges in a spark,as predicted by Maxwell's equations.This convinced physicists that light is indeed an electromagnetic wave. waves travel at speed c=2.998 10m/ s in vacuum.The frequenen avelngtot an electromagnetic wave are relatcd by Av =c 1.10 morized.The Appendix gives the Greel thei s are applied waves dependn and aa e term light to denote any kind of electromagnetic radiation.Wavele gths of visible and ultraviolet radiation were formerly given in a angstroms (A)and are ow given in nano meters (nm): 1nm=109m,1A=10-10m=0.1nm 1.2 In the 1890s,physicists measured the intensity of light at various frequencies emitted by a heated blackbody at a fixed temperature,and did these measurements at sev- eral temperatures.A blackbody is an object that absorbs all light falling on it.A good
2 Chapter 1 | The Schrödinger Equation people have speculated that nanoscience and nanotechnology will bring about the “next industrial revolution.” The rapid increase in computer speed and the development of new methods (such as density functional theory—Section 16.4) of doing molecular calculations have made quantum chemistry a practical tool in all areas of chemistry. Nowadays, several companies sell quantum-chemistry software for doing molecular quantum-chemistry calculations. These programs are designed to be used by all kinds of chemists, not just quantum chemists. Because of the rapidly expanding role of quantum chemistry and related theoretical and computational methods, the American Chemical Society began publication of a new periodical, the Journal of Chemical Theory and Computation, in 2005. “Quantum mechanics . . . underlies nearly all of modern science and technology. It governs the behavior of transistors and integrated circuits . . . and is . . . the basis of modern chemistry and biology” (Stephen Hawking, A Brief History of Time, 1988, Bantam, chap. 4). 1.2 Historical Background of Quantum Mechanics The development of quantum mechanics began in 1900 with Planck’s study of the light emitted by heated solids, so we start by discussing the nature of light. In 1803, Thomas Young gave convincing evidence for the wave nature of light by observing diffraction and interference when light went through two adjacent pinholes. (Diffraction is the bending of a wave around an obstacle. Interference is the combining of two waves of the same frequency to give a wave whose disturbance at each point in space is the algebraic or vector sum of the disturbances at that point resulting from each interfering wave. See any first-year physics text.) In 1864, James Clerk Maxwell published four equations, known as Maxwell’s equations, which unified the laws of electricity and magnetism. Maxwell’s equations predicted that an accelerated electric charge would radiate energy in the form of electromagnetic waves consisting of oscillating electric and magnetic fields. The speed predicted by Maxwell’s equations for these waves turned out to be the same as the experimentally measured speed of light. Maxwell concluded that light is an electromagnetic wave. In 1888, Heinrich Hertz detected radio waves produced by accelerated electric charges in a spark, as predicted by Maxwell’s equations. This convinced physicists that light is indeed an electromagnetic wave. All electromagnetic waves travel at speed c = 2.998 * 108 m/s in vacuum. The frequency n and wavelength l of an electromagnetic wave are related by ln = c (1.1) (Equations that are enclosed in a box should be memorized. The Appendix gives the Greek alphabet.) Various conventional labels are applied to electromagnetic waves depending on their frequency. In order of increasing frequency are radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays, and gamma rays. We shall use the term light to denote any kind of electromagnetic radiation. Wavelengths of visible and ultraviolet radiation were formerly given in angstroms (Å) and are now given in nanometers (nm): 1 nm = 10-9 m, 1 Å = 10-10 m = 0.1 nm (1.2) In the 1890s, physicists measured the intensity of light at various frequencies emitted by a heated blackbody at a fixed temperature, and did these measurements at several temperatures. A blackbody is an object that absorbs all light falling on it. A good
1.2 Historical Background of Quantum Mechanics3 ly rad nd blackbod are empin gy per unit tin nit surf Wien's fo e area by infinit 896b he retical ar ents for the formula were considered unsatisfact In 1899-1900.measurements of blackbody radiation were extended to lower frequen- cies than previously measured,and the low-frequency data showed significant deviations from Wien's formula.These deviations led the physicist Max Planck to propose in October 1900 the following formula:=a(e 1).which was found to give an excellent fit to the data at all frequencies. Having proposed this formula.Planck sought a theoretical justification for it.In December 1900.he presented a theoretical derivation of his equation to the German Physi cal Society.Planck assumed the radiation emitters and absorbers in the blackbody to be harmonically oscillating electric charges("resonators")in equilibrium with electromag- n in a cavi that the tota ergyo nt oI mag wh nd h(Planck's a new in p the ener cac this)Thus the ofeach tha values were allowed for are Planck's the d tha 2mh/c2 and =where k is Boltzmann's constant.By fitting the experimental blackbody curves Planck found=66 x 10-34I.s Planck's work is usually considered to mark the beginning of quantum mechanics However,historians of physics have debated whether Planck in 1900 viewed energy quan- tization as a description of physical reality or as merely a mathematical approximation that allow ed him to obtain the correct blackbody radiation formula.[See O.Darrigol.Cen- taurus,43,219(2001):C.A.Gearhart.Phys.Perspect.,4.170(2002)(available online at employees.csbsju.edu/cgearhart/Planck/PQH.pdf:S.G.Brush,Am.J.Phrys.,70.119 (2002)(www.punsterproductions.com/-sciencehistory/cautious.htm).]The physics histo an World cep revious idea of phy A t contradic on to all th the eg) in the correct blackbody-radiation curves The second application of energy quantization was to the photoelectric effect.In the pho- toelectric effect.lis ght shining on a metal causes emission of electrons.The energy of a wave is proportional to its intensity and is not related to its frequency,so the electromagnetic-wave picture of light leads one to expect that the kinetic energy of an emitted photoelectron would increase as the light intensity increases but would not change as the light frequency changes Instead,one observes that the kinetic energy of an emitted electron is independent of the light's intensity but increases as the light's frequency increases. In 905,Einstein showed that these observations could be explained by regarding light as composed of particlelike entities (called photons),with each photon having an energy Ephoton =h 1.3
1.2 Historical Background of Quantum Mechanics | 3 approximation to a blackbody is a cavity with a tiny hole. In 1896, the physicist Wien proposed the following equation for the dependence of blackbody radiation on light frequency and blackbody temperature: I = an3>ebn>T , where a and b are empirical constants, and I dn is the energy with frequency in the range n to n + dn radiated per unit time and per unit surface area by a blackbody, with dn being an infinitesimal frequency range. Wien’s formula gave a good fit to the blackbody radiation data available in 1896, but his theoretical arguments for the formula were considered unsatisfactory. In 1899–1900, measurements of blackbody radiation were extended to lower frequencies than previously measured, and the low-frequency data showed significant deviations from Wien’s formula. These deviations led the physicist Max Planck to propose in October 1900 the following formula: I = an3> 1ebn>T - 12, which was found to give an excellent fit to the data at all frequencies. Having proposed this formula, Planck sought a theoretical justification for it. In December 1900, he presented a theoretical derivation of his equation to the German Physical Society. Planck assumed the radiation emitters and absorbers in the blackbody to be harmonically oscillating electric charges (“resonators”) in equilibrium with electromagnetic radiation in a cavity. He assumed that the total energy of those resonators whose frequency is n consisted of N indivisible “energy elements,” each of magnitude hn, where N is an integer and h (Planck’s constant) was a new constant in physics. Planck distributed these energy elements among the resonators. In effect, this restricted the energy of each resonator to be a whole-number multiple of hv (although Planck did not explicitly say this). Thus the energy of each resonator was quantized, meaning that only certain discrete values were allowed for a resonator energy. Planck’s theory showed that a = 2ph>c2 and b = h>k, where k is Boltzmann’s constant. By fitting the experimental blackbody curves, Planck found h = 6.6 * 10-34 J # s. Planck’s work is usually considered to mark the beginning of quantum mechanics. However, historians of physics have debated whether Planck in 1900 viewed energy quantization as a description of physical reality or as merely a mathematical approximation that allowed him to obtain the correct blackbody radiation formula. [See O. Darrigol, Centaurus, 43, 219 (2001); C. A. Gearhart, Phys. Perspect., 4, 170 (2002) (available online at employees.csbsju.edu/cgearhart/Planck/PQH.pdf; S. G. Brush, Am. J. Phys., 70, 119 (2002) (www.punsterproductions.com/~sciencehistory/cautious.htm).] The physics historian Kragh noted that “If a revolution occurred in physics in December 1900, nobody seemed to notice it. Planck was no exception, and the importance ascribed to his work is largely a historical reconstruction” (H. Kragh, Physics World, Dec. 2000, p. 31). The concept of energy quantization is in direct contradiction to all previous ideas of physics. According to Newtonian mechanics, the energy of a material body can vary continuously. However, only with the hypothesis of quantized energy does one obtain the correct blackbody-radiation curves. The second application of energy quantization was to the photoelectric effect. In the photoelectric effect, light shining on a metal causes emission of electrons. The energy of a wave is proportional to its intensity and is not related to its frequency, so the electromagnetic-wave picture of light leads one to expect that the kinetic energy of an emitted photoelectron would increase as the light intensity increases but would not change as the light frequency changes. Instead, one observes that the kinetic energy of an emitted electron is independent of the light’s intensity but increases as the light’s frequency increases. In 1905, Einstein showed that these observations could be explained by regarding light as composed of particlelike entities (called photons), with each photon having an energy Ephoton = hn (1.3)
4 Chapter 1 The Schrodinger Equation When an electron in the metal absorbs a photon,part of the absorbed photon energy is used to overcome the forces holding the electron in the metal:the remainder appears as kinetic energy of the electron after it has left the metal.Conservation of energy gives =+T,where is the minimum energy needed by an electron to escape the metal (the metal's work function),and T is the maximum kinetic energy of an emitted electron. 一品enases the photon ery and enceh一 An increase in light intensity at fixed frequencyin ne me ce in the rate but does no netic energy emitted electron. Kragh,a strong】 e ma rld Dec 2000.31 first recognized the essence iclelike behavior in addition to the relike behavior it shows in diffraction In 1907 Einstein applied energy quantization to the vibrations of atoms in a solid ele ment,assuming that each atom's vibra y in each direction ( to be an integer times hvh element.Using statistical mechanics,Einstein derived an expression for the constant- volume heat capacity Cy of the solid.Einstein's equation agreed fairly well with known C.-versus-temperature data for diamond. Now let us consider the structure of matter. In the late nineteenth century,investigations of electric discharge tubes and natu- ral radioactivity showed that atoms and molecules are composed of charged particles Electrons have a negative site in sign to the electron charge and is er tha tuent of atoms,the neutron (discovered in 1932),is uncharged and slightly n the r,and M passed a b alph d the defle ed helium obtained from natural radioactive decay.Most of the alpha particles pa sed thr ough the foil essentially undeflected.but.surprisingly.a few under vent large deflectione me he ing deflected backward.To get large deflections,one needs a very close approach between the charges,so that the Coulombic repulsive force is great.If the positive charge were spread throughout the atom(as J.J.Thomson had proposed in 1904),once the high-energy alpha particle penetrated the atom,the repulsive force would fall off.becoming zero at the center of the atom,according to classical electrostatics.Hence Rutherford concluded that such large deflections could occur only if the positive charge were concentrated in a tiny. heavy nucleus. avy nu us c sisting of neu trons proton e are cles inte acco ing to ont-rang ar fo ons are held es,whic kinetic the ole,by r rom th roperties of atoms and determined byther structure.and so the estion arises as to the nature of the motions and e of the electrons since the nu cleus is much more massive than the electron,we expect the motion of the nucleus to be slight compared with the electrons'motions In 1911.Rutherford proposed his planetary model of the atom in which the elec- trons revolved about the nucleus in various orbits.just as the planets revolve about the sun.However,there is a fundamental difficulty with this model.According to classical
4 Chapter 1 | The Schrödinger Equation When an electron in the metal absorbs a photon, part of the absorbed photon energy is used to overcome the forces holding the electron in the metal; the remainder appears as kinetic energy of the electron after it has left the metal. Conservation of energy gives hn = + T, where is the minimum energy needed by an electron to escape the metal (the metal’s work function), and T is the maximum kinetic energy of an emitted electron. An increase in the light’s frequency n increases the photon energy and hence increases the kinetic energy of the emitted electron. An increase in light intensity at fixed frequency increases the rate at which photons strike the metal and hence increases the rate of emission of electrons, but does not change the kinetic energy of each emitted electron. (According to Kragh, a strong “case can be made that it was Einstein who first recognized the essence of quantum theory”; Kragh, Physics World, Dec. 2000, p. 31.) The photoelectric effect shows that light can exhibit particlelike behavior in addition to the wavelike behavior it shows in diffraction experiments. In 1907, Einstein applied energy quantization to the vibrations of atoms in a solid element, assuming that each atom’s vibrational energy in each direction 1x, y, z2 is restricted to be an integer times hnvib, where the vibrational frequency nvib is characteristic of the element. Using statistical mechanics, Einstein derived an expression for the constantvolume heat capacity CV of the solid. Einstein’s equation agreed fairly well with known CV -versus-temperature data for diamond. Now let us consider the structure of matter. In the late nineteenth century, investigations of electric discharge tubes and natural radioactivity showed that atoms and molecules are composed of charged particles. Electrons have a negative charge. The proton has a positive charge equal in magnitude but opposite in sign to the electron charge and is 1836 times as heavy as the electron. The third constituent of atoms, the neutron (discovered in 1932), is uncharged and slightly heavier than the proton. Starting in 1909, Rutherford, Geiger, and Marsden repeatedly passed a beam of alpha particles through a thin metal foil and observed the deflections of the particles by allowing them to fall on a fluorescent screen. Alpha particles are positively charged helium nuclei obtained from natural radioactive decay. Most of the alpha particles passed through the foil essentially undeflected, but, surprisingly, a few underwent large deflections, some being deflected backward. To get large deflections, one needs a very close approach between the charges, so that the Coulombic repulsive force is great. If the positive charge were spread throughout the atom (as J. J. Thomson had proposed in 1904), once the high-energy alpha particle penetrated the atom, the repulsive force would fall off, becoming zero at the center of the atom, according to classical electrostatics. Hence Rutherford concluded that such large deflections could occur only if the positive charge were concentrated in a tiny, heavy nucleus. An atom contains a tiny (10-13 to 10-12 cm radius), heavy nucleus consisting of neutrons and Z protons, where Z is the atomic number. Outside the nucleus there are Z electrons. The charged particles interact according to Coulomb’s law. (The nucleons are held together in the nucleus by strong, short-range nuclear forces, which will not concern us.) The radius of an atom is about one angstrom, as shown, for example, by results from the kinetic theory of gases. Molecules have more than one nucleus. The chemical properties of atoms and molecules are determined by their electronic structure, and so the question arises as to the nature of the motions and energies of the electrons. Since the nucleus is much more massive than the electron, we expect the motion of the nucleus to be slight compared with the electrons’ motions. In 1911, Rutherford proposed his planetary model of the atom in which the electrons revolved about the nucleus in various orbits, just as the planets revolve about the sun. However, there is a fundamental difficulty with this model. According to classical
1.2 Historical Background of Quantum Mechanics5 clectromagnetic than aclerated charged paricle rdiates energy in the form of waves An el the s at consta in the Ruthe ing.H niral t th Th nd.ac uld to cla ergy A possible way out of this difficulty was pr oposed by niels bobr in 1913 when he ap plied the concept of quantization of energy to the hydrogen atom.Bohr assumed that the energy of the electron in a hydrogen atom was quantized,with the electron constrained to move only on one of a number of allowed circles.When an electron makes a transition from one Bohr orbit to another,a photon of light whose frequency vsatisfies Eupper -Elower =hv 1.49 yh eme and Eower are the energies of the upper and lower from a free (ionized)state to one of the bound orbits emits a photon whose frequency Bohra the c quntro sical mechanics to derive a formula for the hydrogen e got agreemen pts to fi g th led.Mo spect reover,the heo Tthe fail not mical o usin ohr mo sin molecule the el om the of dis that only certain ies of motion a red:the elec nie er Quantization does occur in wave motion-for example,the fundamental and overtone fre. quencies of a violin string.Hence Louis de Broglie suggested in 1923 that the motion of electrons might have a wave aspect;that an electron of mass m and speed would have a wavelength A=品- 1.5 associated with it.where p is the linear momen um.De Broglie arrived at Eq.(1.5)by Eins ing in an gy with p gy of a photor vity,a pe: h s the get pe 、nd din and Germer ex nfirmed de o an ele In 1927.Davissor erimentally e's hy reflecting electrons from metals and observing diffraction effects.In 1932.Stern observed the same effects with helium atoms and hydrogen molecules.thus verifving that the wave effects are not peculiar to electrons,but result from some general law of motion for mi- croscopic particles.Diffraction and interference have been observed with molecules as C分8 e p large as CasH2F C32HisNs molecules can be seen at www.youtube.com/watch?v=vCiOMQIRU7I. Thus electrons behave in some respects like particles and in other respects like waves. We eapparently contradictory "of matter (and of ron be oth a part which is n it a electr description o an electron's impos ible
1.2 Historical Background of Quantum Mechanics | 5 electromagnetic theory, an accelerated charged particle radiates energy in the form of electromagnetic (light) waves. An electron circling the nucleus at constant speed is being accelerated, since the direction of its velocity vector is continually changing. Hence the electrons in the Rutherford model should continually lose energy by radiation and therefore would spiral toward the nucleus. Thus, according to classical (nineteenth-century) physics, the Rutherford atom is unstable and would collapse. A possible way out of this difficulty was proposed by Niels Bohr in 1913, when he applied the concept of quantization of energy to the hydrogen atom. Bohr assumed that the energy of the electron in a hydrogen atom was quantized, with the electron constrained to move only on one of a number of allowed circles. When an electron makes a transition from one Bohr orbit to another, a photon of light whose frequency v satisfies Eupper - Elower = hn (1.4) is absorbed or emitted, where Eupper and Elower are the energies of the upper and lower states (conservation of energy). With the assumption that an electron making a transition from a free (ionized) state to one of the bound orbits emits a photon whose frequency is an integral multiple of one-half the classical frequency of revolution of the electron in the bound orbit, Bohr used classical mechanics to derive a formula for the hydrogenatom energy levels. Using (1.4), he got agreement with the observed hydrogen spectrum. However, attempts to fit the helium spectrum using the Bohr theory failed. Moreover, the theory could not account for chemical bonds in molecules. The failure of the Bohr model arises from the use of classical mechanics to describe the electronic motions in atoms. The evidence of atomic spectra, which show discrete frequencies, indicates that only certain energies of motion are allowed; the electronic energy is quantized. However, classical mechanics allows a continuous range of energies. Quantization does occur in wave motion—for example, the fundamental and overtone frequencies of a violin string. Hence Louis de Broglie suggested in 1923 that the motion of electrons might have a wave aspect; that an electron of mass m and speed v would have a wavelength l = h mv = h p (1.5) associated with it, where p is the linear momentum. De Broglie arrived at Eq. (1.5) by reasoning in analogy with photons. The energy of a photon can be expressed, according to Einstein’s special theory of relativity, as E = pc, where c is the speed of light and p is the photon’s momentum. Using Ephoton = hn, we get pc = hn = hc>l and l = h>p for a photon traveling at speed c. Equation (1.5) is the corresponding equation for an electron. In 1927, Davisson and Germer experimentally confirmed de Broglie’s hypothesis by reflecting electrons from metals and observing diffraction effects. In 1932, Stern observed the same effects with helium atoms and hydrogen molecules, thus verifying that the wave effects are not peculiar to electrons, but result from some general law of motion for microscopic particles. Diffraction and interference have been observed with molecules as large as C48H26F24N8O8 passing through a diffraction grating [T. Juffmann et al., Nat. Nanotechnol., 7, 297 (2012).]. A movie of the buildup of an interference pattern involving C32H18N8 molecules can be seen at www.youtube.com/watch?v=vCiOMQIRU7I. Thus electrons behave in some respects like particles and in other respects like waves. We are faced with the apparently contradictory “wave–particle duality” of matter (and of light). How can an electron be both a particle, which is a localized entity, and a wave, which is nonlocalized? The answer is that an electron is neither a wave nor a particle, but something else. An accurate pictorial description of an electron’s behavior is impossible