xivContents13.3TheWeyl Quantization forR2n26127113.4The"NoGo"Theorem of Groenewold27513.5Exercises27914 The Stone-von Neumann Theorem27914.1 A Heuristic Argument14.2TheExponentiated Commutation Relations28128614.3TheTheorem29214.4The Segal-Bargmann Space14.5301Exercises30515 The WKB Approximation30515.1Introduction15.2The Old Quantum Theory and the Bohr-Sommerfeld306Condition30815.3Classical and Semiclassical Approximations15.4TheWKBApproximationAwayfrom theTurning311Points15.5315The Airy Function and the Connection Formulas32015.6A Rigorous Error Estimate15.7328Other Approaches32915.8Exercises33316 Lie Groups, Lie Algebras, and Representations16.1334Summary16.2335Matrix Lie Groups16.3338Lie Algebras339The MatrixExponential16.434216.5The Lie Algebra of a Matrix Lie Group34416.6Relationships Between Lie Groups and Lie Algebras16.7Finite-Dimensional Representations of Lie Groups350and LieAlgebras16.8New Representations from Old35836016.9Infinite-Dimensional Unitary Representations36316.10 Exercises36717 Angular Momentum and Spin17.1TheRole of Angular Momentum367in QuantumMechanics17.2The Angular Momentum Operators in R336817.3Angular Momentum from the Lie Algebra Point369of View17.4370The Irreducible Representations of so(3)17.5375The IrreducibleRepresentations of SO(3)37617.6Realizing the Representations Inside L?(s2)
xiv Contents 13.3 The Weyl Quantization for R2n . . . . . . . . . . . . . . . 261 13.4 The “No Go” Theorem of Groenewold . . . . . . . . . . . 271 13.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 14 The Stone–von Neumann Theorem 279 14.1 A Heuristic Argument . . . . . . . . . . . . . . . . . . . . 279 14.2 The Exponentiated Commutation Relations . . . . . . . . 281 14.3 The Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 286 14.4 The Segal–Bargmann Space . . . . . . . . . . . . . . . . . 292 14.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 15 The WKB Approximation 305 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 305 15.2 The Old Quantum Theory and the Bohr–Sommerfeld Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 15.3 Classical and Semiclassical Approximations . . . . . . . . . 308 15.4 The WKB Approximation Away from the Turning Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 15.5 The Airy Function and the Connection Formulas . . . . . 315 15.6 A Rigorous Error Estimate . . . . . . . . . . . . . . . . . . 320 15.7 Other Approaches . . . . . . . . . . . . . . . . . . . . . . . 328 15.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 16 Lie Groups, Lie Algebras, and Representations 333 16.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 16.2 Matrix Lie Groups . . . . . . . . . . . . . . . . . . . . . . 335 16.3 Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 338 16.4 The Matrix Exponential . . . . . . . . . . . . . . . . . . . 339 16.5 The Lie Algebra of a Matrix Lie Group . . . . . . . . . . . 342 16.6 Relationships Between Lie Groups and Lie Algebras . . . . 344 16.7 Finite-Dimensional Representations of Lie Groups and Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . 350 16.8 New Representations from Old . . . . . . . . . . . . . . . . 358 16.9 Infinite-Dimensional Unitary Representations . . . . . . . 360 16.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 17 Angular Momentum and Spin 367 17.1 The Role of Angular Momentum in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . 367 17.2 The Angular Momentum Operators in R3 . . . . . . . . . 368 17.3 Angular Momentum from the Lie Algebra Point of View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 17.4 The Irreducible Representations of so(3) . . . . . . . . . . 370 17.5 The Irreducible Representations of SO(3) . . . . . . . . . . 375 17.6 Realizing the Representations Inside L2(S2) . . . . . . . . 376
Contentsxv38017.7Realizing the Representations Inside L?(R3)38317.8Spin17.9Tensor Products of Representations:"Addition of384Angular Momentum"38717.10 Vectors and Vector Operators39017.11 Exercises18Radial Potentials and the Hydrogen Atom39339318.1RadialPotentials39618.2The Hydrogen Atom:Preliminaries39718.3TheBound States of theHydrogen Atom18.4The Runge-Lenz Vector in the Quantum Kepler401Problem40918.5The Role of Spin41018.6Runge-Lenz Calculations41618.7Exercises41919 Systems and Subsystems, Multiple Particles41919.1Introduction19.2421Trace-Class and Hilbert-Schmidt Operators19.3Density Matrices:The General Notion422oftheStateof aQuantumSystem42719.4Modified Axioms for Quantum Mechanics42919.5Composite Systemsand theTensorProduct19.6Multiple Particles:Bosons and Fermions43343519.7"Statistics" and the Pauli Exclusion Principle43819.8Exercises4412o The Path Integral Formulation of Quantum Mechanics44220.1TrotterProduct Formula44420.2Formal Derivation of the Feynman Path Integral20.3The Imaginary-Time Calculation44744820.4The Wiener Measure44920.5The Feynman-Kac Formula20.6451Path Integrals in Quantum Field Theory45320.7Exercises4552l Hamiltonian Mechanics on Manifolds45521.1Calculus on Manifolds21.2459Mechanics on Symplectic Manifolds21.3465Exercises46722Geometric Quantization on Euclidean Space46722.1 Introduction46822.2Prequantization
Contents xv 17.7 Realizing the Representations Inside L2(R3) . . . . . . . . 380 17.8 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 17.9 Tensor Products of Representations: “Addition of Angular Momentum” . . . . . . . . . . . . . . . . . . . . . 384 17.10 Vectors and Vector Operators . . . . . . . . . . . . . . . . 387 17.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 18 Radial Potentials and the Hydrogen Atom 393 18.1 Radial Potentials . . . . . . . . . . . . . . . . . . . . . . . 393 18.2 The Hydrogen Atom: Preliminaries . . . . . . . . . . . . . 396 18.3 The Bound States of the Hydrogen Atom . . . . . . . . . . 397 18.4 The Runge–Lenz Vector in the Quantum Kepler Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 18.5 The Role of Spin . . . . . . . . . . . . . . . . . . . . . . . 409 18.6 Runge–Lenz Calculations . . . . . . . . . . . . . . . . . . . 410 18.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 19 Systems and Subsystems, Multiple Particles 419 19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 419 19.2 Trace-Class and Hilbert–Schmidt Operators . . . . . . . . 421 19.3 Density Matrices: The General Notion of the State of a Quantum System . . . . . . . . . . . . . . 422 19.4 Modified Axioms for Quantum Mechanics . . . . . . . . . 427 19.5 Composite Systems and the Tensor Product . . . . . . . . 429 19.6 Multiple Particles: Bosons and Fermions . . . . . . . . . . 433 19.7 “Statistics” and the Pauli Exclusion Principle . . . . . . . 435 19.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 20 The Path Integral Formulation of Quantum Mechanics 441 20.1 Trotter Product Formula . . . . . . . . . . . . . . . . . . . 442 20.2 Formal Derivation of the Feynman Path Integral . . . . . . 444 20.3 The Imaginary-Time Calculation . . . . . . . . . . . . . . 447 20.4 The Wiener Measure . . . . . . . . . . . . . . . . . . . . . 448 20.5 The Feynman–Kac Formula . . . . . . . . . . . . . . . . . 449 20.6 Path Integrals in Quantum Field Theory . . . . . . . . . . 451 20.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 21 Hamiltonian Mechanics on Manifolds 455 21.1 Calculus on Manifolds . . . . . . . . . . . . . . . . . . . . 455 21.2 Mechanics on Symplectic Manifolds . . . . . . . . . . . . . 459 21.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 22 Geometric Quantization on Euclidean Space 467 22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 467 22.2 Prequantization . . . . . . . . . . . . . . . . . . . . . . . . 468
xviContents22.3ProblemswithPrequantization47247422.4Quantization.47822.5Quantization of Observables48222.6Exercises48323 Geometric Quantization on Manifolds48323.1Introduction23.2485LineBundles and Connections49023.3Prequantization49223.4Polarizations23.5495Quantization Without Half-Forms23.6505Quantization with Half-Forms: The Real Case23.7518Quantization with Half-Forms: The Complex Case23.8521Pairing Maps23.9523Exercises527A Review of Basic Material527A.1TensorProductsof VectorSpaces529A.2MeasureTheory.530A.3Elementary Functional Analysis537A.4HilbertSpacesand OperatorsonThem545ReferencesIndex549
xvi Contents 22.3 Problems with Prequantization . . . . . . . . . . . . . . . 472 22.4 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . 474 22.5 Quantization of Observables . . . . . . . . . . . . . . . . . 478 22.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 23 Geometric Quantization on Manifolds 483 23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 483 23.2 Line Bundles and Connections . . . . . . . . . . . . . . . . 485 23.3 Prequantization . . . . . . . . . . . . . . . . . . . . . . . . 490 23.4 Polarizations . . . . . . . . . . . . . . . . . . . . . . . . . . 492 23.5 Quantization Without Half-Forms . . . . . . . . . . . . . . 495 23.6 Quantization with Half-Forms: The Real Case . . . . . . . 505 23.7 Quantization with Half-Forms: The Complex Case . . . . . 518 23.8 Pairing Maps . . . . . . . . . . . . . . . . . . . . . . . . . 521 23.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 A Review of Basic Material 527 A.1 Tensor Products of Vector Spaces . . . . . . . . . . . . . . 527 A.2 Measure Theory . . . . . . . . . . . . . . . . . . . . . . . . 529 A.3 Elementary Functional Analysis . . . . . . . . . . . . . . . 530 A.4 Hilbert Spaces and Operators on Them . . . . . . . . . . . 537 References 545 Index 549
1The Experimental Origins of QuantumMechanicsQuantum mechanics, with its controversial probabilistic nature and curiousblending of waves and particles, is a very strange theory.It was notinvented because anyone thought this is the way the world should behave,but because various experiments showed that this is the way the worlddoes behave, like it or not. Craig Hogan, director of the Fermilab ParticleAstrophysics Center, put itthis way:No theorist in his right mind would have invented quantummechanics unless forced to by data.1Although thefirst hint of quantum mechanics came in 1900 with Planck'ssolution to the problem of blackbody radiation,the full theory did noemerge until 1925-1926, with Heisenberg's matrix model, Schrodinger'swave model, and Born's statistical interpretation of the wave model.1.1Is Light a Wave or a Particle?1.1.1Newton VersusHuygensBeginning in the late seventeenth century and continuing into the earlyeighteenth century, there was a vigorous debate in the scientific community1Quoted in "Is Space Digital?" by Michael Moyer, Scientific American, February2012, Pp. 3036.1B.C.Hall, Quantum Theory for Mathematicians, GraduateTextsin Mathematics 267, DOI 10.1007/978-1-4614-7116-5_1,Springer Science+Business Media NewYork 2013
1 The Experimental Origins of Quantum Mechanics Quantum mechanics, with its controversial probabilistic nature and curious blending of waves and particles, is a very strange theory. It was not invented because anyone thought this is the way the world should behave, but because various experiments showed that this is the way the world does behave, like it or not. Craig Hogan, director of the Fermilab Particle Astrophysics Center, put it this way: No theorist in his right mind would have invented quantum mechanics unless forced to by data.1 Although the first hint of quantum mechanics came in 1900 with Planck’s solution to the problem of blackbody radiation, the full theory did not emerge until 1925–1926, with Heisenberg’s matrix model, Schr¨odinger’s wave model, and Born’s statistical interpretation of the wave model. 1.1 Is Light a Wave or a Particle? 1.1.1 Newton Versus Huygens Beginning in the late seventeenth century and continuing into the early eighteenth century, there was a vigorous debate in the scientific community 1Quoted in “Is Space Digital?” by Michael Moyer, Scientific American, February 2012, pp. 30–36. B.C. Hall, Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, DOI 10.1007/978-1-4614-7116-5 1, © Springer Science+Business Media New York 2013 1
21.TheExperimental Origins of Quantum Mechanicsover the nature of light.One camp, following the views of IsaacNewton, claimed that light consisted of a group of particles or "corpus-cles." The other camp, led by the Dutch physicist Christiaan Huygens,claimed that light was a wave. Newton argued that only a corpuscular the-ory could account for the observed tendency of light to travel in straightlines.Huygens and others, on the other hand, argued that a wave theorycould explain numerous observed aspects of light, including the bendingor"refraction" of light as it passes from one medium to another, as fromair into water. Newton's reputation was such that his“corpuscular"theoryremained the dominant one until the early nineteenth century.1.1.2The Ascendance of the Wave Theory of LightIn 1804, Thomas Young published two papers describing and explaininghis double-slit experiment. In this experiment, sunlight passes through asmall hole in a piece of cardboard and strikes another piece of cardboardcontaining two small holes.The light then strikes a third piece of cardboard,where the pattern of light may be observed.Young observed“fringes"oralternating regions of high and low intensity for the light.Young believedthat light was a wave and he postulated that these fringes were the resultof interference between the waves emanating from the two holes. Youngdrew an analogy between light and water, where in the case of water,interference is readily observed. If two circular waves of water cross eachother,there will be some points wherea peak of one wave matches up witha trough of another wave, resulting in destructive interference, that is, apartial cancellation between the two waves, resulting in a small amplitudeof the combined wave at that point. At other points, on the other hand, apeak in one wave will line up with a peak in the other, or a trough witha trough. At such points, there is constructive interference, with the resultthat the amplitude of the combined wave is large at that point. The patternof constructive and destructive interference will produce something likeacheckerboard pattern of alternating regions of large and small amplitudesin the combined wave.The dimensions of each region will be roughly onthe order of the wavelength of the individual waves.Based on this analogy with water waves,Young was able to explain theinterference fringes that he observed and to predict the wavelength thatlight must have in order for the specific patterns he observed to occur.Based on his observations, Young claimed that the wavelength of visiblelight ranged from about1/36,000in.(about 700nm)at the red end of thespectrum to about 1/60,000 in. (about 425nm) at the violet end of thespectrum, results that agree with modern measurements.Figure 1.1 shows how circular waves emitted from two different pointsform an interference pattern. One should think of Young's second piece ofcardboard as being at the top of the figure, with holes near the top left and
2 1. The Experimental Origins of Quantum Mechanics over the nature of light. One camp, following the views of Isaac Newton, claimed that light consisted of a group of particles or “corpuscles.” The other camp, led by the Dutch physicist Christiaan Huygens, claimed that light was a wave. Newton argued that only a corpuscular theory could account for the observed tendency of light to travel in straight lines. Huygens and others, on the other hand, argued that a wave theory could explain numerous observed aspects of light, including the bending or “refraction” of light as it passes from one medium to another, as from air into water. Newton’s reputation was such that his “corpuscular” theory remained the dominant one until the early nineteenth century. 1.1.2 The Ascendance of the Wave Theory of Light In 1804, Thomas Young published two papers describing and explaining his double-slit experiment. In this experiment, sunlight passes through a small hole in a piece of cardboard and strikes another piece of cardboard containing two small holes. The light then strikes a third piece of cardboard, where the pattern of light may be observed. Young observed “fringes” or alternating regions of high and low intensity for the light. Young believed that light was a wave and he postulated that these fringes were the result of interference between the waves emanating from the two holes. Young drew an analogy between light and water, where in the case of water, interference is readily observed. If two circular waves of water cross each other, there will be some points where a peak of one wave matches up with a trough of another wave, resulting in destructive interference, that is, a partial cancellation between the two waves, resulting in a small amplitude of the combined wave at that point. At other points, on the other hand, a peak in one wave will line up with a peak in the other, or a trough with a trough. At such points, there is constructive interference, with the result that the amplitude of the combined wave is large at that point. The pattern of constructive and destructive interference will produce something like a checkerboard pattern of alternating regions of large and small amplitudes in the combined wave. The dimensions of each region will be roughly on the order of the wavelength of the individual waves. Based on this analogy with water waves, Young was able to explain the interference fringes that he observed and to predict the wavelength that light must have in order for the specific patterns he observed to occur. Based on his observations, Young claimed that the wavelength of visible light ranged from about 1/36,000 in. (about 700 nm) at the red end of the spectrum to about 1/60,000 in. (about 425 nm) at the violet end of the spectrum, results that agree with modern measurements. Figure 1.1 shows how circular waves emitted from two different points form an interference pattern. One should think of Young’s second piece of cardboard as being at the top of the figure, with holes near the top left and