Invitation To Quantum Mechanics
iInvitation toQuantum MechanicsDaniel F. Styer
iiInvitation to Quantum MechanicsDaniel F. StyerSchiffer Professor of Physics, Oberlin Collegecopyright c 20 January 2021Daniel F. StyerThe copyright holder grants the freedom to copy, modify, convey, adapt,and/or redistribute this work under the terms of the Creative CommonsAttribution Share Alike 4.0 International License. A copy of that license isavailable at lcode.You may freely download this book in pdf format onToQM.It is formatted to print nicely on either A4 or U.S. Letter paper. You mayalso purchase a printed and bound copy from World Scientific PublishingCompany. In neither case does the author receive monetary gain from yourdownload/purchase: it is reward enough for him that you want to explorequantum mechanics.
Love all God’s creation, the whole and every grain of sand in it.Love the stars, the trees, the thunderstorms, the atoms.The more you love, the more you will grow curious.The more you grow curious, the more you will question.The more you question, the more you will uncover.The more you uncover, the more you will love.And so at last you will come to love the entire universe with anagile and resilient love founded upon facts and understanding.— This improvisation by Dan Styer was inspired bythe first sentence, which appears inFyodor Dostoyevsky’s The Brothers Karamazov.iii
ivDedicated to Linda Ong Styer, adventurer
ContentsSynoptic Contents1Welcome31.92.“Something Isn’t Quite Right”1.1Light in thermal equilibrium: Blackbody radiation . . . .101.2Photoelectric effect . . . . . . . . . . . . . . . . . . . . . .231.3Wave character of electrons . . . . . . . . . . . . . . . . .301.4How does an electron behave? . . . . . . . . . . . . . . . .351.5Quantization of atomic energies . . . . . . . . . . . . . . .361.6Quantization of magnetic moment . . . . . . . . . . . . .42What Is Quantum Mechanics About?472.1Quantization . . . . . . . . . . . . . . . . . . . . . . . . .472.2Interference . . . . . . . . . . . . . . . . . . . . . . . . . .592.3Aharonov-Bohm effect . . . . . . . . . . . . . . . . . . . .672.4Light on the atoms . . . . . . . . . . . . . . . . . . . . . .702.5Entanglement . . . . . . . . . . . . . . . . . . . . . . . . .722.6Quantum cryptography. . . . . . . . . . . . . . . . . . .842.7What is a qubit? . . . . . . . . . . . . . . . . . . . . . . .88v
vi3.4.ContentsForging Mathematical Tools893.1What is a quantal state? . . . . . . . . . . . . . . . . . . .893.2Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . .913.3Reversal-conjugation relation . . . . . . . . . . . . . . . .993.4Establishing a phase convention . . . . . . . . . . . . . . . 1013.5How can I specify a quantal state? . . . . . . . . . . . . . 1033.6States for entangled systems . . . . . . . . . . . . . . . . . 1123.7Are states “real”? . . . . . . . . . . . . . . . . . . . . . . . 1163.8What is a qubit? . . . . . . . . . . . . . . . . . . . . . . . 116The Quantum Mechanics of Position1194.1Probability and probability density: One particle in onedimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.2Probability amplitude . . . . . . . . . . . . . . . . . . . . 1244.3How does wavefunction change with time? . . . . . . . . . 1264.4Wavefunction: Two particles . . . . . . . . . . . . . . . . . 1274.5Solving the Schrödinger time evolution equation for theinfinite square well . . . . . . . . . . . . . . . . . . . . . . 1314.6What did we learn by solving the Schrödinger time evolution equation for the infinite square well? . . . . . . . . . 1394.7Other potentials4.8Energy loss . . . . . . . . . . . . . . . . . . . . . . . . . . 1524.9Mean values . . . . . . . . . . . . . . . . . . . . . . . . . . 153. . . . . . . . . . . . . . . . . . . . . . . 1494.10 The classical limit of quantum mechanics. . . . . . . . . 1574.11 Transitions induced by light . . . . . . . . . . . . . . . . . 1664.12 Position plus spin . . . . . . . . . . . . . . . . . . . . . . . 172
Contents5.6.7.8.viiSolving the Energy Eigenproblem1775.1Sketching energy eigenfunctions . . . . . . . . . . . . . . . 1785.2Scaled quantities . . . . . . . . . . . . . . . . . . . . . . . 1985.3Numerical solution of the energy eigenproblem . . . . . . 201Identical Particles2056.1Two or three identical particles . . . . . . . . . . . . . . . 2056.2Symmetrization and antisymmetrization . . . . . . . . . . 2086.3Consequences of the Pauli principle . . . . . . . . . . . . . 2146.4Consequences of the Pauli principle for product states . . 2176.5Energy states for two identical, noninteracting particles . 2176.6Spin plus space, two electrons . . . . . . . . . . . . . . . . 2196.7Spin plus space, three electrons, ground state . . . . . . . 225Atoms2297.1Central potentials in two dimensions . . . . . . . . . . . . 2297.2Central potentials in three dimensions . . . . . . . . . . . 2367.3The hydrogen atom . . . . . . . . . . . . . . . . . . . . . . 2387.4The helium atom . . . . . . . . . . . . . . . . . . . . . . . 2477.5The lithium atom . . . . . . . . . . . . . . . . . . . . . . . 2527.6All other atoms . . . . . . . . . . . . . . . . . . . . . . . . 2537.7The periodic table . . . . . . . . . . . . . . . . . . . . . . 256The Vistas Open to Us259Appendix ASignificant Figures263Appendix BDimensions269
viiiContentsAppendix CComplex Arithmetic277Appendix DProblem-Solving Tips and Techniques279Appendix ECatalog of Misconceptions283Index285
Synoptic ContentsWelcomeWhat is quantum mechanics and why should I care about it?1. “Something Isn’t Quite Right”Historical experiments show that classical mechanics is flawed.2. What Is Quantum Mechanics About?If classical mechanics is wrong, then what is right? We explore, in thecontext of modern experiments with qubits, the atomic phenomena thatquantum mechanics needs to explain.3. Forging Mathematical ToolsWe build a framework for the quantum mechanics of qubits, using amathematical tool called “amplitude”.4. The Quantum Mechanics of PositionThe framework, built to treat qubits, extends to treat continuum positionas well. Energy plays a central role here.1
2Synoptic Contents5. Solving the Energy EigenproblemSince energy plays a central role, we devote a chapter to solving suchproblems. We find that solving particular problems strengthens ourconceptual understanding, and that conceptual understanding strengthensour skill in solving particular problems.6. Identical ParticlesThis surprisingly subtle topic deserves a chapter of its own.7. AtomsWe apply our new knowledge to physical (rather than model) systems.8. The Vistas Open to UsThis book is an invitation. Where might you and quantum mechanics traveltogether?
WelcomeWhy would anyone want to study quantum mechanics?Starting in the year 1900, physicists exploring the newly discovered atomfound that the atomic world of electrons and protons is not just smaller thanour familiar world of trees, balls, and automobiles, it is also fundamentallydifferent in character. Objects in the atomic world obey different rules fromthose obeyed by a tossed ball or an orbiting planet. These atomic rules areso different from the familiar rules of everyday physics, so counterintuitiveand unexpected, that it took more than 25 years of intense research touncover them.But it is really only since the year 1990 that physicists have come toappreciate that the rules of the atomic world (now called “quantum mechanics”) are not just different from the everyday rules (now called “classicalmechanics”). The atomic rules are also far richer. The atomic rules providefor phenomena like particle interference and entanglement that are simplyabsent from the everyday world. Every phenomenon of classical mechanicsis also present in quantum mechanics, but the quantum world provides formany additional phenomena.Here’s an analogy: Some films are in black-and-white and some are incolor. It does not malign any black-and-white film to say that a color filmhas more possibilities, more richness. In fact, black-and-white films aresimply one category of color films, because black and white are both colors.Anyone moving from the world of only black-and-white to the world of coloris opening up the door to a new world — a world ripe with new possibilitiesand new expression — without closing the door to the old world.This same flood of richness and freshness comes from entering the quantum world. It is a difficult world to enter, because we humans have no expe3
4Welcomerience, 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 appliedto slow things (that is, with speeds much less than light speed c). The speedat which the classical approximation becomes legitimate depends upon the
Welcome5accuracy 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.
6WelcomeWhat you can expect from this bookThis book introduces quantum mechanics at the second-year American undergraduate level. It assumes the reader knows about classical forces, potential energy functions, and the simple harmonic oscillator. The readershould know that wavelength is represented by λ, frequency by f , and thatfor a wave moving at speed c, λf c. S/he needs to know the meaningand significance of “standard deviation”. Turning to mathematics, it assumes the reader knows about complex numbers (see appendix C) and dotproducts, knows the difference between an ordinary and a partial derivative, and can solve simple ordinary differential equations. It assumes thatthe reader understands phrases like “orthonormal basis representation of aposition vector”.This is a book about physics, not mathematics. The word “physics”derives from the Greek word for “nature”, so the emphasis lies in nature,not in the mathematics we use to describe nature. Thus the book startswith experiments about nature, then builds mathematical machinery todescribe nature, and finally applies the machinery to atoms, where theunderstanding of both nature and machinery is deepened.The book never abandons its focus on nature. It provides a balanced,interwoven treatment of concepts, techniques, 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.I do not provide summary lists of key ideas and difficult-to-rememberconcepts and equations. That’s because an equation that I find easy toremember might be hard for you to remember. I recommend instead that
Welcome7you write out for yourself, in your own words, a summary of the ideas andequations that you consider most important and most difficult to remember.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.Teaching tipsMost physics departments offer a second-year course titled ModernPhysics. The topics in this course vary widely from institution to institution: special relativity and elementary quantum mechanics are staples, butthe course might also cover classical waves, thermodynamics and elementary statistical mechanics, descriptive atomic, molecular, and solid statephysics. No textbook could cover all this variety, nor should any textbooktry: instead each institution should provide a mix of topics appropriatefor its own students. This book is devoted only to quantum mechanics atthe level of a Modern Physics course. You will want to add it to othermaterials for the other topics in your own particular course.In chapters 2 and 3, a surprising amount of student difficulty comes fromnothing more than getting straight which Stern-Gerlach analyzer is orientedin which direction. I recommend that you mark up some cardboard boxesto look like analyzers and analyzer loops, and use them as demonstrationsduring your classes.Chapter 5 presents two techniques for solving the energy eigenproblem:one informal and one numerical. I discuss the first in class and assignthe second for reading, because the first benefits from a lot of blackboardsketching, erasing, resketching, and gesturing. But your own prioritiesmight differ from mine, so you might take the opposite tack.This text spends a lot of time on concepts before applying those conceptsto atoms. Atoms are mathematically intense, and it pays to get the conceptsstraight first. If we jumped directly into atoms, that mathematical intensitywould completely obscure the conceptual issues. Some people like it thatway, because they don’t want to face the conceptual issues.
8WelcomeAcknowledgmentsI learned quantum mechanics from stellar teachers. My high school chemistry teacher Frank Dugan introduced me not only to quantum mechanicsbut to the precept that science involves hard, fulfilling work in additionto dreams and imagination. When I was an undergraduate, John Bocciohelped mold my understanding of quantum mechanics, and also molded theshape of my life. In graduate school N. David Mermin, Vinay Ambegaokar,Neil Ashcroft, and Michael Peskin pushed me without mercy but pushedme in the direction of understanding and away from the mind-numbing attitude of “shut up and calculate”. My debt to my thesis adviser, MichaelFisher, is incalculable. I’ve been inspired by research lectures from TonyLeggett, Jürg Fröhlich, Jennifer and Lincoln Chayes, Shelly Goldstein, andChris Fuchs, among others.I have taught quantum mechanics to thousands of students from thegeneral audience level through advanced undergraduates. Their questions,confusions, triumphs, and despairs have infused my own understanding ofthe discipline. I cannot name them all, but I would be remiss if I did notthank my former students Paul Kimoto, Gail Welsh, K. Tabetha Hole, GaryFelder, Sarah Clemmens, Dahyeon Lee, and Noah Morris.In the fall 2020 semester I taught Modern Physics at Oberlin Collegeusing a draft of this textbook. I received helpful corrections and suggestionsfrom several students, but especially from Ilana Meisler. Thank you.My scientific prose style was developed by Michael Fisher and N. DavidMermin. In particular this book’s structure of “first lay out the phenomena(chapters 1 and 2), then build mathematical tools to describe those phenomena” echos the structure of Fisher’s 1964 essay “The Nature of CriticalPoints”. I have also absorbed lessons in writing from John McPhee, Maurice Forrester, and Terry Tempest Williams. My teaching style has beeninfluenced especially by Mark Heald, Tony French, Edwin Taylor, ArnoldArons, and Robert H. Romer.Shelley Kronzek, my editor at World Scientific, kept faith in this projectthrough multiple delays. The book was skillfully copyedited by MatthewAbbate (who was also my undergraduate roommate and who served as apotential guardian to my children).
Chapter 1“Something Isn’t Quite Right”We 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 f couldtake on only the energies ofE nhf,where n 0, 1, 2, 3, . .
quantum mechanics relativistic mechanics size small big Finally, is there a framework that applies to situations that are both fast and small? There is: it is called \relativistic quantum mechanics" and is closely related to \quantum eld theory". Ordinary non-relativistic quan-tum mechanics is a good approximation for relativistic quantum mechanics
1. Introduction - Wave Mechanics 2. Fundamental Concepts of Quantum Mechanics 3. Quantum Dynamics 4. Angular Momentum 5. Approximation Methods 6. Symmetry in Quantum Mechanics 7. Theory of chemical bonding 8. Scattering Theory 9. Relativistic Quantum Mechanics Suggested Reading: J.J. Sakurai, Modern Quantum Mechanics, Benjamin/Cummings 1985
1. Quantum bits In quantum computing, a qubit or quantum bit is the basic unit of quantum information—the quantum version of the classical binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics.
An excellent way to ease yourself into quantum mechanics, with uniformly clear expla-nations. For this course, it covers both approximation methods and scattering. Shankar, Principles of Quantum Mechanics James Binney and David Skinner, The Physics of Quantum Mechanics Weinberg, Lectures on Quantum Mechanics
Quantum Mechanics 6 The subject of most of this book is the quantum mechanics of systems with a small number of degrees of freedom. The book is a mix of descriptions of quantum mechanics itself, of the general properties of systems described by quantum mechanics, and of techniques for describing their behavior.
mechanics, it is no less important to understand that classical mechanics is just an approximation to quantum mechanics. Traditional introductions to quantum mechanics tend to neglect this task and leave students with two independent worlds, classical and quantum. At every stage we try to explain how classical physics emerges from quantum .
EhrenfestEhrenfest s’s Theorem The expectation value of quantum mechanics followsThe expectation value of quantum mechanics follows the equation of motion of classical mechanics. In classical mechanics In quantum mechanics, See Reed 4.5 for the proof. Av
According to the quantum model, an electron can be given a name with the use of quantum numbers. Four types of quantum numbers are used in this; Principle quantum number, n Angular momentum quantum number, I Magnetic quantum number, m l Spin quantum number, m s The principle quantum
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