The Physics Of Quantum Mechanics - Oberlin College

2WelcomeThis same flood of richness and freshness comes from entering the quantum world. It is a difficult world to enter, because we humans have no experience, no intuition, no expectations about this world. Even our language,invented by people living in the everyday world, has no words for the newquantal phenomena — just as a language among a race of the color-blindwould have no word for “red”.Reading this book is not easy: it is like a color-blind student learningabout color from a color-blind teacher. The book is just one long argument,building up the structure of a world that we can explore not through touchor through sight or through scent, but only through logic. Those willing tofollow and to challenge the logic, to open their minds to a new world, willfind themselves richly rewarded.The place of quantum mechanics in natureQuantum mechanics is the framework for describing and analyzing smallthings, like atoms and nuclei. Quantum mechanics also applies to bigthings, like baseballs and galaxies, but when applied to big things, certain approximations become legitimate: taken together, these are calledthe classical approximation to quantum mechanics, and the result is thefamiliar classical mechanics.Quantum mechanics is not only less familiar and less intuitive thanclassical mechanics; it is also harder than classical mechanics. So wheneverthe classical approximation is sufficiently accurate, we would be foolish notto use it. This leads some to develop the misimpression that quantummechanics applies to small things, while classical mechanics applies to bigthings. No. Quantum mechanics applies to all sizes, but classical mechanicsis a good approximation to quantum mechanics when it is applied to bigthings.For what size is the classical approximation good enough? That dependson the accuracy desired. The higher the accuracy demanded, the more situations will require full quantal treatment rather than approximate classicaltreatment. But as a rule of thumb, something as big as a DNA strand isalmost always treated classically, not quantum mechanically.This situation is analogous to the relationship between relativistic mechanics and classical mechanics. Relativity applies always, but classicalmechanics is a good approximation to relativistic mechanics when applied

Welcome3to slow things (that is, with speeds much less than light speed c). The speedat which the classical approximation becomes legitimate depends upon theaccuracy demanded, but as a rule of thumb particles moving less than aquarter of light speed are treated classically.The difference between the quantal case and the relativistic case is thatwhile relativistic mechanics is less familiar, less comforting, and less expected than classical mechanics, it is no more intricate than classical mechanics. Quantum mechanics, in contrast, is less familiar, less comforting,less expected, and more intricate than classical mechanics. This intricacymakes quantum mechanics harder than classical mechanics, yes, but alsoricher, more textured, more nuanced. Whether to curse or celebrate thisintricacy is your nics0smallsizebigFinally, is there a framework that applies to situations that are both fastand small? There is: it is called “relativistic quantum mechanics” and isclosely related to “quantum field theory”. Ordinary non-relativistic quantum mechanics is a good approximation for relativistic quantum mechanicswhen applied to slow things. Relativistic mechanics is a good approximation for relativistic quantum mechanics when applied to big things. Andclassical mechanics is a good approximation for relativistic quantum mechanics when applied to big, slow things.

4WelcomeWhat you can expect from this bookThis book introduces quantum mechanics at the third- or fourth-year American undergraduate level. It assumes the reader knows about. . . .This is a book about physics, not about mathematics. The word“physics” derives from the Greek word for “nature”, so the emphasis lies innature, not in the mathematics we use to describe nature. Thus the bookstarts with experiments about nature, then builds mathematical machineryto describe nature, then erects a formalism (“postulates”), and then moveson to applications, where the formalism is applied to nature and where theunderstanding of both nature and formalism is deepened.The book never abandons its focus on nature. It provides a balanced,interwoven treatment of concepts, formalism, and applications so that eachthread reinforces the other. There are many problems at many levels ofdifficulty, but no problem is there just for “make-work”: each has a “moralto the story”. Some problems are essential to the logical development ofthe subject: these are labeled (unsurprisingly) “essential”. Other problemspromote learning far better than simple reading can: these are labeled“recommended”. Sample problems build both mathematical technique andphysical insight.The book does not merely convey correct ideas, it also refutes misconceptions. Just to get started, I list the most important and most perniciousmisconceptions about quantum mechanics: (a) An electron has a positionbut you don’t know what it is. (b) The only states are energy states. (c) Thewavefunction ψ( x, t) is “out there” in space and you could reach out andtouch it if only your fingers were sufficiently sensitive.The object of the biographical footnotes in this book is twofold: First, topresent the briefest of outlines of the subject’s historical development, lestanyone get the misimpression that quantum mechanics arose fully formed,like Aphrodite from sea foam. Second, to show that the founders of quantum mechanics were not inaccessible giants, but people with foibles andstrengths, with interests both inside and outside of physics, just like youand me.

Chapter 1What is Quantum Mechanics About?1.1QuantizationWe are used to things that vary continuously: An oven can take on anytemperature, a recipe might call for any quantity of flour, a child can grow toa range of heights. If I told you that an oven might take on the temperatureof 172.1 C or 181.7 C, but that a temperature of 173.8 C was physicallyimpossible, you would laugh in my face.So you can imagine the surprise of physicists on 14 December 1900,when Max Planck announced that certain features of blackbody radiation(that is, of light in thermal equilibrium) could be explained by assumingthat the energy of the light could not take on any value, but only certaindiscrete values. Specifically, Planck found that light of frequency ω couldtake on only the energies ofE ω(n 12 ),where n 0, 1, 2, 3, . . .,(1.1)and where the constant (now called the “reduced Planck constant”) is 1.054 571 817 10 34 J s.(1.2)(I use modern terminology and the current value for , rather than theterminology and value used by Planck in 1900.)That is, light of frequency ω can have an energy of 3.5 ω, and it canhave an energy of 4.5 ω, but it is physically impossible for this light to havean energy of 3.8 ω. Any numerical quantity that can take on only discretevalues like this is called “quantized”. By contrast, a numerical quantitythat can take on any value is called “continuous”.The photoelectric effect supplies additional evidence that the energy oflight comes only in discrete values. And if the energy of light comes in5

6What is Quantum Mechanics About?discrete values, then it’s a good guess that the energy of an atom comes indiscrete values too. This good guess was confirmed through investigations ofatomic spectra (where energy goes into or out of an atom via absorption oremission of light) and through the Franck–Hertz experiment (where energygoes into or out of an atom via collisions).Furthermore, if the energy of an atom comes in discrete values, thenit’s a good guess that other properties of an atom — such as its magneticmoment — also take on only discrete values. The theme of this book isthat these good guesses have all proved to be correct.The story of Planck’s1 discovery is a fascinating one, but it’s a difficultand elaborate story because it involves not just quantization, but also thermal equilibrium and electromagnetic radiation. The story of the discoveryof atomic energy quantization is just as fascinating, but again fraught withintricacies. In an effort to remove the extraneous and dive deep to the heartof the matter, we focus on the magnetic moment of an atom. We will, to theextent possible, do a quantum-mechanical treatment of an atom’s magneticmoment while maintaining a classical treatment of all other aspects — suchas its energy and momentum and position. (In chapter 6, “The QuantumMechanics of Position”, we take up a quantum-mechanical treatment ofposition, momentum, and energy.)1.1.1The Stern-Gerlach experimentAn electric current flowing in a loop produces a magnetic moment, so itmakes sense that the electron orbiting (or whatever it does) an atomicnucleus would produce a magnetic moment for that atom. And of course, italso makes sense that physicists would be itching to measure that magneticmoment.It is not difficult to measure the magnetic moment of, say, a scoutcompass. Place the magnetic compass needle in a known magnetic fieldand measure the torque that acts to align the needle with the field. You1 Max Karl Ernst Ludwig Planck (1858–1947) was a German theoretical physicist particularly interested in thermodynamics and radiation. Concerning his greatest discovery,the introduction of quantization into physics, he wrote, “I can characterize the whole procedure as an act of desperation, since, by nature I am peaceable and opposed to doubtfuladventures.” [Letter from Planck to R.W. Wood, 7 October 1931, quoted in J. Mehraand H. Rechenberg, The Historical Development of Quantum Theory (Springer–Verlag,New York, 1982) volume 1, page 49.]

1.1. Quantization7will need to measure an angle and you might need to look up a formula inyour magnetism textbook, but there is no fundamental difficulty.Measuring the magnetic moment of an atom is a different matter. Youcan’t even see an atom, so you can’t watch it twist in a magnetic field like acompass needle. Furthermore, because the atom is very small, you expectthe associated magnetic moment to be very small, and hence very hard tomeasure. The technical difficulties are immense.These difficulties must have deterred but certainly did not stop OttoStern and Walter Gerlach.2 They realized that the twisting of a magneticmoment in a uniform magnetic field could not be observed for atomic-sizedmagnets, and also that the moment would experience zero net force. Butthey also realized that a magnetic moment in a non-uniform magnetic fieldwould experience a net force, and that this force could be used to measurethe magnetic moment. Bz6µ A classical magnetic moment in a non-uniform magnetic field. thatA classical magnetic moment µ , situated in a magnetic field Bpoints in the z direction and increases in magnitude in the z direction, issubject to a force B,(1.3)µz zwhere µz is the z-component of the magnetic moment or, in other words,the projection of µ on the z axis. (If this is not obvious to you, then workproblem 1.1, “Force on a classical magnetic moment”, on page 9.)2 Otto Stern (1888–1969) was a Polish-German-Jewish physicist who made contributionsto both theory and experiment. He left Germany for the United States in 1933 uponthe Nazi ascension to power. Walter Gerlach (1889–1979) was a German experimentalphysicist. During the Second World War he led the physics section of the Reich ResearchCouncil and for a time directed the German effort to build a nuclear bomb.

8What is Quantum Mechanics About?Stern and Gerlach used this fact to measure the z-component of themagnetic moment of an atom. First, they heated silver in an electric “oven”.The vaporized silver atoms emerged from a pinhole in one side of the oven,and then passed through a non-uniform magnetic field. At the far side ofthe field the atoms struck and stuck to a glass plate. The entire apparatushad to be sealed within a good vacuum, so that collisions with nitrogenmolecules would not push the silver atoms around. The deflection of anatom away from straight-line motion is proportional to the magnetic force,and hence proportional to the projection µz . In this ingenious way, Sternand Gerlach could measure the z-component of the magnetic moment of anatom even though any single atom is invisible.Before reading on, pause and think about what results you would expectfrom this experiment.Here are the results that I expect: I expect that an atom which happensto enter the field with magnetic moment pointing straight up (in the zdirection) will experience a large upward force. Hence it will move upwardand stick high up on the glass-plate detector. I expect that an atom whichhappens to enter with magnetic moment pointing straight down (in the zdirection) will experience a large downward force, and hence will stick fardown on the glass plate. I expect that an atom entering with magneticmoment tilted upward, but not straight upward, will move upward butnot as far up as the straight-up atoms, and the mirror image for an atomentering with magnetic moment tilted downward. I expect that an atomentering with horizontal magnetic moment will experience a net force ofzero, so it will pass through the non-uniform field undeflected.Furthermore, I expect that when a silver atom emerges from the ovensource, its magnetic moment will be oriented randomly — as likely to pointin one direction as in any other. There is only one way to point straight up,so I expect that very few atoms will stick high on the glass plate. There aremany ways to point horizontally, so I expect many atoms to pass throughundeflected. There is only one way to point straight down, so I expect veryfew atoms to stick far down on the glass plate.3In summary, I expect that atoms would leave the magnetic field in any ofa range of deflections: a very few with large positive deflection, more with a3 To be specific, this reasoning suggests that the number of atoms with moment tiltedat angle θ relative to the z direction is proportional to sin θ, where θ ranges from 0 to180 . You might want to prove this to yourself, but we’ll never use this result so don’tfeel compelled.

1.1. Quantization9small positive deflection, a lot with no deflection, some with a small negativedeflection, and a very few with large negative deflection. This continuity ofdeflections reflects a continuity of magnetic moment projections.In fact, however, this is not what happens at all! The projection µzdoes not take on a continuous range of values. Instead, it is quantized andtakes on only two values, one positive and one negative. Those two valuesare called µz µB where µB , the so-called “Bohr magneton”, has themeasured value ofµB 9.274 010 078 10 24 J/T,(1.4)with an uncertainty of 3 in the last decimal digit.Distribution of µzExpected:Actual:µzµz µB00 µBThe Stern-Gerlach experiment was initially performed with silver atomsbut has been repeated with many other types of atoms. When nitrogen isused, the projection µz takes on one of the four quantized values of 3µB , µB , µB , or 3µB . When sulfur is used, it takes on one of the fivequantized values of 4µB , 2µB , 0, 2µB , and 4µB . For no atom do thevalues of µz take on the broad continuum of my classical expectation. Forall atoms, the projection µz is quantized.Problems1.1 Force on a classical magnetic momentThe force on a classical magnetic moment is most easily calculatedusing “magnetic charge fiction”: Consider the magnetic moment

10What is Quantum Mechanics About?to consist of two “magnetic charges” of magnitude m and m,separated by the position vector d running from m to m. The magnetic moment is then µ md.a. Use B for the magnitude of the magnetic field at m, andB for the magnitude of the magnetic field at m. Show thatthe net force on the magnetic moment is in the z direction withmagnitude mB mB . Show that to highb. Use dz for the z-component of the vector d.accuracy Bdz . zSurely, for distances of atomic scale, this accuracy is morethan adequate.c. Derive expression (1.3) for the force on a magnetic moment.B B 1.1.2The conundrum of projectionsI would expect the projections µz of a silver atom to take on a continuousrange of values. But in fact, these values are quantized: Whenever µzis measured, it turns out to be either µB or µB , and never anythingelse. This is counterintuitive and unexpected, but we can live with thecounterintuitive and unexpected — it happens all the time in politics.However, this fact of quantization appears to result in a logical contradiction, because there are many possible axes upon which the magneticmoment can be projected. The figures on the next page make it clear thatit is impossible for any vector to have a projection of either µB on allaxes!

1.1. Quantization11Because if the projection of µ on the z axis is µB . . .z µBµ . . . then the projection of µ on this second axis must be more than µB . . .zµ . . . while the projection of µ on this third axis must be less than µB .zµ Whenever we measure the magnetic moment, projected onto any axis,the result is either µB or µB . Yet is it impossible for the projectionof any classical arrow on all axes to be either µB or µB ! This seeming

12What is Quantum Mechanics About?contradiction is called “the conundrum of projections”. We can live withthe counterintuitive, the unexpected, the strange, but we cannot live witha logical contradiction. How can we resolve it?The resolution comes not from meditating on the question, but fromexperimenting about it. Let us actually measure the projection on oneaxis, and then on a second. To do this easily, we modify the Stern-Gerlachapparatus and package it into a box called a “Stern-Gerlach analyzer”. Thisbox consists of a Stern-Gerlach apparatus followed by “pipes” that channelthe outgoing atoms into horizontal paths.4 This chapter treats only silveratoms, so we use analyzers with two exit ports.packaged intoAAn atom enters a vertical analyzer through the single hole on the left.If it exits through the upper hole on the right (the “ port”) then theoutgoing atom has µz µB . If it exits through the lower hole on theright (the “ port”) then the outgoing atom has µz µB .µz µBµz µB4 In general, the “pipes” will manipulate the atoms through electromagnetic fields, notthrough touching. One way way to make such “pipes” is to insert a second Stern-Gerlachapparatus, oriented upside-down relative to the first. The atoms with µz µB , whichhad experienced an upward force in the first half, will experience an equal downwardforce in the second half, and the net impulse delivered will be zero. But whatever theirmanner of construction, the pipes must not change the magnetic moment of an atompassing through them.

1.1. Quantization1.1.313Two vertical analyzersIn order to check the operation of our analyzers, we do preliminary experiments. Atoms are fed into a vertical analyzer. Any atom exiting from the port is then channeled into a second vertical analyzer. That atom exitsfrom the port of the second analyzer. This makes sense: the atom hadµz µB when exiting the first analyzer, and the second analyzer confirmsthat it has µz µB .allµz µBnoneµz µB(ignore these)Furthermore, if an atom exiting from the port of the first analyzeris channeled into a second vertical analyzer, then that atom exits from the port of the second analyzer.1.1.4One vertical and one upside-down analyzerAtoms are fed into a vertical analyzer. Any atom exiting from the port isthen channeled into a second analyzer, but this analyzer is oriented upsidedown. What happens? If the projection on an upward-pointing axis is µB(that is, µz µB ), then the projection on a downward-pointing axis is µB (we write this as µ( z) µB ). So I expect that these atoms willemerge

The place of quantum mechanics in nature Quantum mechanics is the framework for describing and analyzing small things, like atoms and nuclei. Quantum mechanics also applies to big things, like baseballs and galaxies, but when applied to big things, cer-tain approximations become legitimate: taken together, these are called