131.3SchrodingerandHeisenbergbdFIGURE1.4.Four images from the 1989 experiment at Hitachi showing theimpact of individual electrons gradually building up to form an interference pat-tern.Image by Akira Tonomura and Wikimedia Commons user Belsazar.Fileis licensed under the Creative Commons Attribution-Share Alike 3.0 Unportedlicense.are shot toward the slits.Rather, it is the distinctive interference patternthat is surprising, with rapid variations in the pattern of electron strikesover short distances, including regions where almost no electron strikesoccur. (Compare Fig.1.4 to Fig.1.2.) Note also that in the experiment, theelectrons are widely separated, so that there is never more than one electronin the apparatus at any one time. Thus, the electrons cannot interfere withoneanother; rather, each electron interferes with itself.Figure 1.4 showsresults from the Hitachi experiment,with the number of observed electronsincreasing from about 150 in the first image to 160,000 in the last image.1.3 Schrodinger and HeisenbergIn1925,Werner Heisenbergproposed a modelofquantum mechanics basedon treating the position and momentum of the particle as, essentially.matricesof size oox oo.Actually.Heisenberghimselfwasnotfamiliar withthe theory of matrices.which was not a standard part of the mathematicaleducation ofphysicistsatthetime.Nevertheless,hehad quantities of theform jk and pjk (where j and k each vary over all integers), which wecan recognize as matrices, as well as expressions such as , jipuk, whichwe can recognize as a matrix product. After Heisenberg explained his the-ory to Max Born, Born recognized the connection of Heisenberg's formulasto matrix theory and made the matrix point of view explicit, in a paper
1.3 Schr¨odinger and Heisenberg 13 FIGURE 1.4. Four images from the 1989 experiment at Hitachi showing the impact of individual electrons gradually building up to form an interference pattern. Image by Akira Tonomura and Wikimedia Commons user Belsazar. File is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license. are shot toward the slits. Rather, it is the distinctive interference pattern that is surprising, with rapid variations in the pattern of electron strikes over short distances, including regions where almost no electron strikes occur. (Compare Fig. 1.4 to Fig. 1.2.) Note also that in the experiment, the electrons are widely separated, so that there is never more than one electron in the apparatus at any one time. Thus, the electrons cannot interfere with one another; rather, each electron interferes with itself. Figure 1.4 shows results from the Hitachi experiment, with the number of observed electrons increasing from about 150 in the first image to 160,000 in the last image. 1.3 Schr¨odinger and Heisenberg In 1925, Werner Heisenberg proposed a model of quantum mechanics based on treating the position and momentum of the particle as, essentially, matrices of size ∞×∞. Actually, Heisenberg himself was not familiar with the theory of matrices, which was not a standard part of the mathematical education of physicists at the time. Nevertheless, he had quantities of the form xjk and pjk (where j and k each vary over all integers), which we can recognize as matrices, as well as expressions such as l xjlplk, which we can recognize as a matrix product. After Heisenberg explained his theory to Max Born, Born recognized the connection of Heisenberg’s formulas to matrix theory and made the matrix point of view explicit, in a paper
141.TheExperimental Origins of QuantumMechanicscoauthored by Born and his assistant, Pascual Jordan. Born, Heisenberg.and Jordan then all published apaper together elaborating upon their the-ory. The papers of Heisenberg, of Born and Jordan, and of Born, Heisen-berg,and Jordan all appeared in 1925.Heisenberg received the1932NobelPrize in physics (actually awarded in 1933) for his work.Born's exclusionfrom this prize was controversial, and may have been influenced by Jordan'sconnections with the Nazi party in Germany. (Heisenberg's own work forthe Nazis during World War II was also a source of much controversy afterthe war.)In any case, Born was awarded the Nobel Prize in physics in1954 for his work on the statistical interpretation of quantum mechanics(Sect.1.4).Meanwhile,in 1926,Erwin Schrodingerpublished four remarkablepapersin which he proposed a wave theory of quantum mechanics, along the linesof thede Broglie hypothesis. In these papers, Schrodinger described howthewaves evolve over time and showed that the energy levels of,for example,the hydrogen atom could be understood as eigenalues of a certain oper-ator.(See Chap.18 for the computation for hydrogen.)Schrodinger alsoshowed that the Heisenberg-Born-Jordan matrix model could be incorpo-rated into the wave theory, thus showing that the matrix theory and thewavetheorywereequivalent(seeSect.3.8).Thisbookdescribesthemath-ematical structure ofquantum mechanics in essentially theform proposedby Schrodinger in 1926.Schrodinger shared the1933 Nobel Prize in physicswithPaul Dirac.1.4AMatterofInterpretationAlthough Schrodinger's 1926 papers gave the correct mathematical descrip-tion of quantum mechanics (as it is generally accepted today), he did notprovide a widely accepted interpretation of the theory. That task fell toBorn,who in a 1926 paper proposed that the"wave function"(as thewaveappearing in the Schrodinger equation is generally called) should be inter-preted statistically,thatis,asdeterminingtheprobabilitiesforobservationsof thesystem.Over time, Born'sstatistical approach developed intotheCopenhagen interpretation of quantum mechanics. Under this interpreta-tion, the wave function of the system is not directly observable. Rather, merely determines the probability of observing a particular result.In particular, if is properly normalized, then the quantity b(r)? isthe probability distribution for the position of theparticle.Even if itselfis spread out over a large region in space, any measurement of the positionof the particle will show that the particle is located at a single point, justas we seefor the electronsin the two-slit experiment in Fig.1.4.Thus,a
14 1. The Experimental Origins of Quantum Mechanics coauthored by Born and his assistant, Pascual Jordan. Born, Heisenberg, and Jordan then all published a paper together elaborating upon their theory. The papers of Heisenberg, of Born and Jordan, and of Born, Heisenberg, and Jordan all appeared in 1925. Heisenberg received the 1932 Nobel Prize in physics (actually awarded in 1933) for his work. Born’s exclusion from this prize was controversial, and may have been influenced by Jordan’s connections with the Nazi party in Germany. (Heisenberg’s own work for the Nazis during World War II was also a source of much controversy after the war.) In any case, Born was awarded the Nobel Prize in physics in 1954 for his work on the statistical interpretation of quantum mechanics (Sect. 1.4). Meanwhile, in 1926, Erwin Schr¨odinger published four remarkable papers in which he proposed a wave theory of quantum mechanics, along the lines of the de Broglie hypothesis. In these papers, Schr¨odinger described how the waves evolve over time and showed that the energy levels of, for example, the hydrogen atom could be understood as eigenvalues of a certain operator. (See Chap. 18 for the computation for hydrogen.) Schr¨odinger also showed that the Heisenberg–Born–Jordan matrix model could be incorporated into the wave theory, thus showing that the matrix theory and the wave theory were equivalent (see Sect. 3.8). This book describes the mathematical structure of quantum mechanics in essentially the form proposed by Schr¨odinger in 1926. Schr¨odinger shared the 1933 Nobel Prize in physics with Paul Dirac. 1.4 A Matter of Interpretation Although Schr¨odinger’s 1926 papers gave the correct mathematical description of quantum mechanics (as it is generally accepted today), he did not provide a widely accepted interpretation of the theory. That task fell to Born, who in a 1926 paper proposed that the “wave function” (as the wave appearing in the Schr¨odinger equation is generally called) should be interpreted statistically, that is, as determining the probabilities for observations of the system. Over time, Born’s statistical approach developed into the Copenhagen interpretation of quantum mechanics. Under this interpretation, the wave function ψ of the system is not directly observable. Rather, ψ merely determines the probability of observing a particular result. In particular, if ψ is properly normalized, then the quantity |ψ(x)| 2 is the probability distribution for the position of the particle. Even if ψ itself is spread out over a large region in space, any measurement of the position of the particle will show that the particle is located at a single point, just as we see for the electrons in the two-slit experiment in Fig. 1.4. Thus, a
151.4AMatterof Interpretationmeasurement of a particle's position does not show the particle“smearedout" over a large region of space, even if the wave function b is smearedoutoveralargeregion.Consider, for example, how Born's interpretation of the Schrodingerequation would play out in the context of the Hitachi double-slit exper-iment depicted in Fig.1.4.Born would say that each electron has a wavefunction that evolves in time according to the Schrodinger equation (anequation of wave type).Each particle's wave function, then, will propa-gate through the slits in a manner similar to that pictured in Fig.1.l. Ifthere is a screen at the bottom of Fig.1.1, then the electron will hit thescreen at a single point, even though the wave function is very spread out.Thewavefunctiondoesnotdeterminewheretheparticlehitsthescreen;itmerely determines the probabilities for where the particle hits the screen.Ifa whole sequence of electrons passes through the slits, one after the other,over time a probability distribution will emerge,determined by the squareof the magnitude of the wave function, which is shown in Fig.1.2. Thus,the probability distribution of electrons, as seen from a large number ofelectrons as in Fig.1.4, shows wavelike interference patterns, even thougheach individual electron strikes the screen at a single point.It is essential to the theory that the wave function b(r) itself is not theprobabilitydensityforthelocationoftheparticle.Rather,theprobabilitydensity is(r)/.The difference is crucial, because probability densitiesare intrinsically positive and thus do not exhibit destructive interference.The wave function itself, however, is complex-valued, and the real andimaginary parts of the wave function take on both positive and negativevalues.whichcan interfere constructively or destructively.The part of thewave function passing through the first slit, for example, can interfere withthepart of thewavefunctionpassingthrough the second slit.Only afterthis interference has taken place do we take the magnitudesquared of thewave function to obtain the probability distribution, which will, therefore,show the sorts of peaks and valleys we see in Fig.1.2.Born's introduction of a probabilistic element into the interpretation ofquantum mechanics was-and to some extent still is-controversial.Ein-stein, for example, is often quoted as saying something along the lines of,"God does not play at dice with the universe."Einstein expressed the samesentiment in various ways over the years. His earliest known statement tothis effect wasinaletter toBorninDecember 1926,inwhichhesaid.Quantum mechanics is certainly imposing.But an inner voicetells me that it is not yet the real thing. The theory says a lot,but does not really bring us any closer to the secret of the"oldone." I, at any rate, am convinced that He does not throw dice.Many other physicists and philosophers have questioned the probabilisticinterpretation of quantum mechanics,andhavesoughtalternatives,suchas"hidden variable"theories.Nevertheless,the Copenhagen interpretation
1.4 A Matter of Interpretation 15 measurement of a particle’s position does not show the particle “smeared out” over a large region of space, even if the wave function ψ is smeared out over a large region. Consider, for example, how Born’s interpretation of the Schr¨odinger equation would play out in the context of the Hitachi double-slit experiment depicted in Fig. 1.4. Born would say that each electron has a wave function that evolves in time according to the Schr¨odinger equation (an equation of wave type). Each particle’s wave function, then, will propagate through the slits in a manner similar to that pictured in Fig. 1.1. If there is a screen at the bottom of Fig. 1.1, then the electron will hit the screen at a single point, even though the wave function is very spread out. The wave function does not determine where the particle hits the screen; it merely determines the probabilities for where the particle hits the screen. If a whole sequence of electrons passes through the slits, one after the other, over time a probability distribution will emerge, determined by the square of the magnitude of the wave function, which is shown in Fig. 1.2. Thus, the probability distribution of electrons, as seen from a large number of electrons as in Fig. 1.4, shows wavelike interference patterns, even though each individual electron strikes the screen at a single point. It is essential to the theory that the wave function ψ(x) itself is not the probability density for the location of the particle. Rather, the probability density is |ψ(x)| 2 . The difference is crucial, because probability densities are intrinsically positive and thus do not exhibit destructive interference. The wave function itself, however, is complex-valued, and the real and imaginary parts of the wave function take on both positive and negative values, which can interfere constructively or destructively. The part of the wave function passing through the first slit, for example, can interfere with the part of the wave function passing through the second slit. Only after this interference has taken place do we take the magnitude squared of the wave function to obtain the probability distribution, which will, therefore, show the sorts of peaks and valleys we see in Fig. 1.2. Born’s introduction of a probabilistic element into the interpretation of quantum mechanics was—and to some extent still is—controversial. Einstein, for example, is often quoted as saying something along the lines of, “God does not play at dice with the universe.” Einstein expressed the same sentiment in various ways over the years. His earliest known statement to this effect was in a letter to Born in December 1926, in which he said, Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the “old one.” I, at any rate, am convinced that He does not throw dice. Many other physicists and philosophers have questioned the probabilistic interpretation of quantum mechanics, and have sought alternatives, such as “hidden variable” theories. Nevertheless, the Copenhagen interpretation
161.The Experimental Origins of Quantum Mechanicsof quantum mechanics, essentially as proposed by Born in 1926, remainsthe standard one. This book resolutely avoids all controversies surround-ing the interpretation of quantum mechanics. Chapter 3, for example,presents the standard statistical interpretation of the theory without ques-tion. The book may nevertheless be of use to the more philosophicallyminded reader, in that one must learn something of quantum mechanicsbefore delving into the (often highly technical) discussions about its inter-pretation.1.5Exercises1. Beginning with the formula for the sum of a geometric series, usedifferentiation to obtain theidentity+e-AEne(1-e-A)2n=02.In Planck's model of blackbody radiation, theenergy in a given fre-quency w of electromagnetic radiation is distributed randomly overall numbers of the form nhw, where n=0,1,2,...Specifically, thelikelihood of finding energy nhw is postulated to be1-βnhwp(E=nhw)-1Z=1-e-βhwwhere Z is a normalization constant, which is chosen so that the sumover n of the probabilities is 1. Here β = 1/(kBT), where T is thetemperature and kB is Boltzmann's constant.The erpected value ofthe energy,denoted (E),isdefined to beA1E(nhw)e-Bnhw(E) =1=0(a)UsingExercise1, showthathw(E)=eBhu-1(b) Show that (E) behaves like 1/β = kT for small w, but that(E)decays exponentiallyas w tends to infinityNote: In applying the above calculation to blackbody radiation, onemust also take into account the number of modes having frequency
16 1. The Experimental Origins of Quantum Mechanics of quantum mechanics, essentially as proposed by Born in 1926, remains the standard one. This book resolutely avoids all controversies surrounding the interpretation of quantum mechanics. Chapter 3, for example, presents the standard statistical interpretation of the theory without question. The book may nevertheless be of use to the more philosophically minded reader, in that one must learn something of quantum mechanics before delving into the (often highly technical) discussions about its interpretation. 1.5 Exercises 1. Beginning with the formula for the sum of a geometric series, use differentiation to obtain the identity ∞ n=0 ne−An = e−A (1 − e−A)2 . 2. In Planck’s model of blackbody radiation, the energy in a given frequency ω of electromagnetic radiation is distributed randomly over all numbers of the form nω, where n = 0, 1, 2,. Specifically, the likelihood of finding energy nω is postulated to be p(E = nω) = 1 Z e−βnω, Z = 1 1 − e−βω where Z is a normalization constant, which is chosen so that the sum over n of the probabilities is 1. Here β = 1/(kBT ), where T is the temperature and kB is Boltzmann’s constant. The expected value of the energy, denoted E, is defined to be E = 1 Z ∞ n=0 (nω)e−βnω. (a) Using Exercise 1, show that E = ω eβω − 1 . (b) Show that E behaves like 1/β = kBT for small ω, but that E decays exponentially as ω tends to infinity. Note: In applying the above calculation to blackbody radiation, one must also take into account the number of modes having frequency
171.5 Exercisesin a given range, say between wo and wo + e. The exact number ofsuch frequencies depends on the shape of the cavity,but according toWeyl's law, this number will be approximately proportional to ewg forlargevalues ofwo.Thus,the amount of energy per unit offrequency ishw3C(1.7)eBhw-1'where C is a constant involving the volume of the cavity and thespeed of light. The relation (1.7) is known as Planck's law.3. In classical mechanics, the kinetic energy of an electron is meu /2,where u is the magnitude of the velocity. Meanwhile, the potentialenergy associated with the force law (1.3) is V(r) = -Q?/r, sincedV/dr = F. Show that if the particle is moving in a circular orbitwith radius rn given by (1.5), then the total energy (kinetic pluspotential) of the particle is En, as given in (1.1)
1.5 Exercises 17 in a given range, say between ω0 and ω0 + ε. The exact number of such frequencies depends on the shape of the cavity, but according to Weyl’s law, this number will be approximately proportional to εω2 0 for large values of ω0. Thus, the amount of energy per unit of frequency is C ω3 eβω − 1 , (1.7) where C is a constant involving the volume of the cavity and the speed of light. The relation (1.7) is known as Planck’s law. 3. In classical mechanics, the kinetic energy of an electron is mev2/2, where v is the magnitude of the velocity. Meanwhile, the potential energy associated with the force law (1.3) is V (r) = −Q2/r, since dV /dr = F. Show that if the particle is moving in a circular orbit with radius rn given by (1.5), then the total energy (kinetic plus potential) of the particle is En, as given in (1.1).