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Notes on Quantum Mechanics K. Schulten Department of Physics and Beckman Institute University of Illinois at Urbana–Champaign 405 N. Mathews Street, Urbana, IL 61801 USA (April 18, 2000) Preface i Preface The following notes introduce Quantum Mechanics at an advanced level addressing students of Physics, Mathematics, Chemistry and Electrical Engineering. The aim is to put mathematical concepts and tech- niques like the path integral, algebraic techniques, Lie algebras and representation theory at the readers disposal. For this purpose we attempt to motivate the various physical and mathematical concepts as well as provide detailed derivations and complete sample calculations. We have made every effort to include in the derivations all assumptions and all mathematical steps implied, avoiding omission of supposedly ‘trivial’ information. Much of the author’s writing effort went into a web of cross references accompanying the mathe- matical derivations such that the intelligent and diligent reader should be able to follow the text with relative ease, in particular, also when mathematically difficult material is presented. In fact, the author’s driving force has been his desire to pave the reader’s way into territories unchartered previously in most introduc- tory textbooks, since few practitioners feel obliged to ease access to their field. Also the author embraced enthusiastically the potential of the T E X typesetting language to enhance the presentation of equations as to make the logical pattern behind the mathematics as transparent as possible. Any suggestion to improve the text in the respects mentioned are most welcome. It is obvious, that even though these notes attempt to serve the reader as much as was possible for the author, the main effort to follow the text and to master the material is left to the reader. The notes start out in Section 1 with a brief review of Classical Mechanics in the Lagrange formulation and build on this to introduce in Section 2 Quantum Mechanics in the closely related path integral formulation. In Section 3 the Schr¨odinger equation is derived and used as an alternative description of continuous quantum systems. Section 4 is devoted to a detailed presentation of the harmonic oscillator, introducing algebraic techniques and comparing their use with more conventional mathematical procedures. In Section 5 we introduce the presentation theory of the 3-dimensional rotation group and the group SU(2) presenting Lie algebra and Lie group techniques and applying the methods to the theory of angular momentum, of the spin of single particles and of angular momenta and spins of composite systems. In Section 6 we present the theory of many–boson and many–fermion systems in a formulation exploiting the algebra of the associated creation and annihilation operators. Section 7 provides an introduction to Relativistic Quantum Mechanics which builds on the representation theory of the Lorentz group and its complex relative Sl(2, C). This section makes a strong effort to introduce Lorentz–invariant field equations systematically, rather than relying mainly on a heuristic amalgam of Classical Special Relativity and Quantum Mechanics. The notes are in a stage of continuing development, various sections, e.g., on the semiclassical approximation, on the Hilbert space structure of Quantum Mechanics, on scattering theory, on perturbation theory, on Stochastic Quantum Mechanics, and on the group theory of elementary particles will be added as well as the existing sections expanded. However, at the present stage the notes, for the topics covered, should be complete enough to serve the reader. The author would like to thank Markus van Almsick and Heichi Chan for help with these notes. The author is also indebted to his department and to his University; their motivated students and their inspiring atmosphere made teaching a worthwhile effort and a great pleasure. These notes were produced entirely on a Macintosh II computer using the T E X typesetting system, Textures, Mathematica and Adobe Illustrator. Klaus Schulten University of Illinois at Urbana–Champaign August 1991 ii Preface Contents 1 Lagrangian Mechanics 1 1.1 Basics of Variational Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Symmetry Properties in Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . . . 7 2 Quantum Mechanical Path Integral 11 2.1 The Double Slit Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Axioms for Quantum Mechanical Description of Single Particle . . . . . . . . . . . . 11 2.3 How to Evaluate the Path Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Propagator for a Free Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.5 Propagator for a Quadratic Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.6 Wave Packet Moving in Homogeneous Force Field . . . . . . . . . . . . . . . . . . . 25 2.7 Stationary States of the Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . 34 3 The Schr¨odinger Equation 51 3.1 Derivation of the Schr¨odinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.3 Particle Flux and Schr¨odinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.4 Solution of the Free Particle Schr¨odinger Equation . . . . . . . . . . . . . . . . . . . 57 3.5 Particle in One-Dimensional Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.6 Particle in Three-Dimensional Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4 Linear Harmonic Oscillator 73 4.1 Creation and Annihilation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.2 Ground State of the Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.3 Excited States of the Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . 78 4.4 Propagator for the Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.5 Working with Ladder Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.6 Momentum Representation for the Harmonic Oscillator . . . . . . . . . . . . . . . . 88 4.7 Quasi-Classical States of the Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . 90 5 Theory of Angular Momentum and Spin 97 5.1 Matrix Representation of the group SO(3) . . . . . . . . . . . . . . . . . . . . . . . . 97 5.2 Function space representation of the group SO(3) . . . . . . . . . . . . . . . . . . . . 104 5.3 Angular Momentum Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 iii iv Contents 5.4 Angular Momentum Eigenstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.5 Irreducible Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.6 Wigner Rotation Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.7 Spin 1 2 and the group SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.8 Generators and Rotation Matrices of SU(2) . . . . . . . . . . . . . . . . . . . . . . . 128 5.9 Constructing Spin States with Larger Quantum Numbers Through Spinor Operators 129 5.10 Algebraic Properties of Spinor Operators . . . . . . . . . . . . . . . . . . . . . . . . 131 5.11 Evaluation of the Elements d j m m (β) of the Wigner Rotation Matrix . . . . . . . . . 138 5.12 Mapping of SU(2) onto SO(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6 Quantum Mechanical Addition of Angular Momenta and Spin 141 6.1 Clebsch-Gordan Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.2 Construction of Clebsch-Gordan Coefficients . . . . . . . . . . . . . . . . . . . . . . . 147 6.3 Explicit Expression for the Clebsch–Gordan Coefficients . . . . . . . . . . . . . . . . 151 6.4 Symmetries of the Clebsch-Gordan Coefficients . . . . . . . . . . . . . . . . . . . . . 160 6.5 Example: Spin–Orbital Angular Momentum States . . . . . . . . . . . . . . . . . . 163 6.6 The 3j–Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 6.7 Tensor Operators and Wigner-Eckart Theorem . . . . . . . . . . . . . . . . . . . . . 176 6.8 Wigner-Eckart Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 7 Motion in Spherically Symmetric Potentials 183 7.1 Radial Schr¨odinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 7.2 Free Particle Described in Spherical Coordinates . . . . . . . . . . . . . . . . . . . . 188 8 Interaction of Charged Particles with Electromagnetic Radiation 203 8.1 Description of the Classical Electromagnetic Field / Separation of Longitudinal and Transverse Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 8.2 Planar Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 8.3 Hamilton Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 8.4 Electron in a Stationary Homogeneous Magnetic Field . . . . . . . . . . . . . . . . . 210 8.5 Time-Dependent Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 215 8.6 Perturbations due to Electromagnetic Radiation . . . . . . . . . . . . . . . . . . . . 220 8.7 One-Photon Absorption and Emission in Atoms . . . . . . . . . . . . . . . . . . . . . 225 8.8 Two-Photon Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 9 Many–Particle Systems 239 9.1 Permutation Symmetry of Bosons and Fermions . . . . . . . . . . . . . . . . . . . . . 239 9.2 Operators of 2nd Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 9.3 One– and Two–Particle Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 9.4 Independent-Particle Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 9.5 Self-Consistent Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 9.6 Self-Consistent Field Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 9.7 Properties of the SCF Ground State . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 9.8 Mean Field Theory for Macroscopic Systems . . . . . . . . . . . . . . . . . . . . . . 272 Contents v 10 Relativistic Quantum Mechanics 285 10.1 Natural Representation of the Lorentz Group . . . . . . . . . . . . . . . . . . . . . . 286 10.2 Scalars, 4–Vectors and Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 10.3 Relativistic Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 10.4 Function Space Representation of Lorentz Group . . . . . . . . . . . . . . . . . . . . 300 10.5 Klein–Gordon Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 10.6 Klein–Gordon Equation for Particles in an Electromagnetic Field . . . . . . . . . . . 307 10.7 The Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 10.8 Lorentz Invariance of the Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . . 317 10.9 Solutions of the Free Particle Dirac Equation . . . . . . . . . . . . . . . . . . . . . . 322 10.10Dirac Particles in Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . 333 11 Spinor Formulation of Relativistic Quantum Mechanics 351 11.1 The Lorentz Transformation of the Dirac Bispinor . . . . . . . . . . . . . . . . . . . 351 11.2 Relationship Between the Lie Groups SL(2,C) and SO(3,1) . . . . . . . . . . . . . . 354 11.3 Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 11.4 Spinor Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 11.5 Lorentz Invariant Field Equations in Spinor Form . . . . . . . . . . . . . . . . . . . . 369 12 Symmetries in Physics: Isospin and the Eightfold Way 371 12.1 Symmetry and Degeneracies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 12.2 Isospin and the SU(2) flavor symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 375 12.3 The Eightfold Way and the flavor SU(3) symmetry . . . . . . . . . . . . . . . . . . 380 vi Contents Chapter 1 Lagrangian Mechanics Our introduction to Quantum Mechanics will be based on its correspondence to Classical Mechanics. For this purpose we will review the relevant concepts of Classical Mechanics. An important concept is that the equations of motion of Classical Mechanics can be based on a variational principle, namely, that along a path describing classical motion the action integral assumes a minimal value (Hamiltonian Principle of Least Action). 1.1 Basics of Variational Calculus The derivation of the Principle of Least Action requires the tools of the calculus of variation which we will provide now. Definition: A functional S[ ] is a map S[ ] : F → R ; F = {q(t); q : [t 0 , t 1 ] ⊂ R → R M ; q(t) differentiable} (1.1) from a space F of vector-valued functions q(t) onto the real numbers. q(t) is called the trajec- tory of a system of M degrees of freedom described by the configurational coordinates q(t) = (q 1 (t), q 2 (t), . . . q M (t)). In case of N classical particles holds M = 3N, i.e., there are 3N configurational coordinates, namely, the position coordinates of the particles in any kind of coordianate system, often in the Cartesian coordinate system. It is important to note at the outset that for the description of a classical system it will be necessary to provide information q(t) as well as d dt q(t). The latter is the velocity vector of the system. Definition: A functional S[ ] is differentiable, if for any q(t) ∈ F and δq(t) ∈ F where F = {δq(t); δq(t) ∈ F, |δq(t)| < , | d dt δq(t)| < , ∀t, t ∈ [t 0 , t 1 ] ⊂ R} (1.2) a functional δS[ ·, ·] exists with the properties (i) S[q(t) + δq(t)] = S[q(t)] + δS[q(t), δq(t)] + O( 2 ) (ii) δS[q(t), δq(t)] is linear in δq(t). (1.3) δS[ ·, ·] is called the differential of S[ ]. The linearity property above implies δS[q(t), α 1 δq 1 (t) + α 2 δq 2 (t)] = α 1 δS[q(t), δq 1 (t)] + α 2 δS[q(t), δq 2 (t)] . (1.4) 1 2 Lagrangian Mechanics Note: δq(t) describes small variations around the trajectory q(t), i.e. q(t) + δq(t) is a ‘slightly’ different trajectory than q(t). We will later often assume that only variations of a trajectory q(t) are permitted for which δq(t 0 ) = 0 and δq(t 1 ) = 0 holds, i.e., at the ends of the time interval of the trajectories the variations vanish. It is also important to appreciate that δS[ ·, ·] in conventional differential calculus does not corre- spond to a differentiated function, but rather to a differential of the function which is simply the differentiated function multiplied by the differential increment of the variable, e.g., df = df dx dx or, in case of a function of M variables, df = M j=1 ∂f ∂x j dx j . We will now consider a particular class of functionals S[ ] which are expressed through an integral over the the interval [t 0 , t 1 ] where the integrand is a function L(q(t), d dt q(t), t) of the configuration vector q(t), the velocity vector d dt q(t) and time t. We focus on such functionals because they play a central role in the so-called action integrals of Classical Mechanics. In the following we will often use the notation for velocities and other time derivatives d dt q(t) = ˙ q(t) and dx j dt = ˙x j . Theorem: Let S[q(t)] = t 1 t 0 dt L(q(t), ˙ q(t), t) (1.5) where L( ·, ·, ·) is a function differentiable in its three arguments. It holds δS[q(t), δq(t)] = t 1 t 0 dt M j=1 ∂L ∂q j − d dt ∂L ∂ ˙q j δq j (t) + M j=1 ∂L ∂ ˙q j δq j (t) t 1 t 0 . (1.6) For a proof we can use conventional differential calculus since the functional (1.6) is expressed in terms of ‘normal’ functions. We attempt to evaluate S[q(t) + δq(t)] = t 1 t 0 dt L(q(t) + δq(t), ˙ q(t) + δ ˙ q(t), t) (1.7) through Taylor expansion and identification of terms linear in δq j (t), equating these terms with δS[q(t), δq(t)]. For this purpose we consider L(q(t) + δq(t), ˙ q(t) + δ ˙ q(t), t) = L(q(t), ˙ q(t), t) + M j=1 ∂L ∂q j δq j + ∂L ∂ ˙q j δ ˙q j + O( 2 ) (1.8) We note then using d dt f(t)g(t) = ˙ f(t)g(t) + f(t)˙g(t) ∂L ∂ ˙q j δ ˙q j = d dt ∂L ∂ ˙q j δq j − d dt ∂L ∂ ˙q j δq j . (1.9) This yields for S[q(t) + δq(t)] S[q(t)] + t 1 t 0 dt M j=1 ∂L ∂q j − d dt ∂L ∂ ˙q j δq j + t 1 t 0 dt M j=1 d dt ∂L ∂ ˙q j δq j + O( 2 ) (1.10) From this follows (1.6) immediately. 1.1: Variational Calculus 3 We now consider the question for which functions the functionals of the type (1.5) assume extreme values. For this purpose we define Definition: An extremal of a differentiable functional S[ ] is a function q e (t) with the property δS[q e (t), δq(t)] = 0 for all δq(t) ∈ F . (1.11) The extremals q e (t) can be identified through a condition which provides a suitable differential equation for this purpose. This condition is stated in the following theorem. Theorem: Euler–Lagrange Condition For the functional defined through (1.5), it holds in case δq(t 0 ) = δq(t 1 ) = 0 that q e (t) is an extremal, if and only if it satisfies the conditions (j = 1, 2, . . . , M ) d dt ∂L ∂ ˙q j − ∂L ∂q j = 0 (1.12) The proof of this theorem is based on the property Lemma: If for a continuous function f(t) f : [t 0 , t 1 ] ⊂ R → R (1.13) holds t 1 t 0 dt f (t)h(t) = 0 (1.14) for any continuous function h(t) ∈ F with h(t 0 ) = h(t 1 ) = 0, then f(t) ≡ 0 on [t 0 , t 1 ]. (1.15) We will not provide a proof for this Lemma. The proof of the above theorem starts from (1.6) which reads in the present case δS[q(t), δq(t)] = t 1 t 0 dt M j=1 ∂L ∂q j − d dt ∂L ∂ ˙q j δq j (t) . (1.16) This property holds for any δq j with δq(t) ∈ F . According to the Lemma above follows then (1.12) for j = 1, 2, . . . M. On the other side, from (1.12) for j = 1, 2, . . . M and δq j (t 0 ) = δq j (t 1 ) = 0 follows according to (1.16) the property δS[q e (t), ·] ≡ 0 and, hence, the above theorem. An Example As an application of the above rules of the variational calculus we like to prove the well-known result that a straight line in R is the shortest connection (geodesics) between two points (x 1 , y 1 ) and (x 2 , y 2 ). Let us assume that the two points are connected by the path y(x), y(x 1 ) = y 1 , y(x 2 ) = y 2 . The length of such path can be determined starting from the fact that the incremental length ds in going from point (x, y(x)) to (x + dx, y(x + dx)) is ds = (dx) 2 + ( dy dx dx) 2 = dx 1 + ( dy dx ) 2 . (1.17) [...]... j ,k= 1 dyN −1 exp i j ,k= 1 yj ajk yk d +∞ dy1 · · · −∞ N −1 yj bjk yk = (iπ)d det(bjk ) (2.35) 1 2 (2.36) which holds for a d-dimensional, real, symmetric matrix (bjk ) and det(bjk ) = 0 In order to complete the evaluation of (2.32) we split off the factor 2 mN in the definition (2.34) of (ajk ) defining a new matrix (Ajk ) through ajk = m 2 Ajk (2.37) N Using det(ajk ) = m 2 N N −1 det(Ajk )... · b allows one to rewrite the e constant of motion e · (r × mv) which can be identified as the component of the angular momentum ˆ mr × v in the e direction It was, of course, to be expected that this is the constant of motion ˆ The important result to be remembered for later considerations of symmetry transformations in the context of Quantum Mechanics is that it is sufficient to know the consequences... that one adds the total time derivative of a function K( r, t) the Lagrangian This term is d K K K K K K(r, t) = x1 + ˙ x2 + ˙ x3 + ˙ = ( K) · v + dt ∂x1 ∂x2 ∂x3 ∂t ∂t (1.43) To prove this theorem we determine the action integral corresponding to the transformed Lagrangian t1 S [q(t)] = t1 ˙ dtL (q, q, t) = t0 = S[q(t)] + q K( q, t) c ˙ dtL(q, q, t) + t0 t1 q K( q, t) c t1 t0 (1.44) t0 Since the condition... that the required solution xcl (t) involves a solution of the Euler–Lagrange equations for boundary conditions which are different from those conventionally encountered in Classical Mechanics where usually a solution for initial conditions xcl (t0 ) = x0 and xcl (t0 ) = v0 are determined ˙ 2.6 Wave Packet Moving in Homogeneous Force Field We want to consider now the motion of a quantum mechanical particle,... Hamiltonian Principle of Least Action we inspect the Euler–Lagrange conditions associated with the action integral defined through (1.22, 1.23) These conditions read in the present case ∂L d ∂L ∂U d − = 0 → − − (mj qj ) = 0 ˙ (1.24) ∂qj dt ∂ qj ˙ ∂qj dt which are obviously equivalent to the Newtonian equations of motion Particle Moving in an Electromagnetic Field We will now consider the Newtonian equations... rotations of coordinates around the center The following description of spatial symmetry is important in two respects, for the connection between invariance properties and constants of motion, which has an important analogy in Quantum Mechanics, and for the introduction of infinitesimal transformations which will provide a crucial method for the study of symmetry in Quantum Mechanics The transformations... transformations we consider are the most simple kind, the reason being that our interest lies in achieving familiarity with the principles (just mentioned above ) of symmetry properties rather than in providing a general tool in the context of Classical Mechanics The transformations considered are specified in the following definition Definition: Infinitesimal One-Parameter Coordinate Transformations A one-parameter... Gauge Transformation of Lagrangian The equation of motion (Euler–Lagrange conditions) of a classical mechanical system are unaffected by the following transformation of its Lagrangian d q ˙ ˙ K( q, t) L (q, q, t) = L(q, q, t) + dt c (1.42) This transformation is termed gauge transformation The factor q has been introduced to make this c transformation equivalent to the gauge transformation (1.32, 1.33)... the system We have used here the notation corresponding to single particle motion, however, the property holds for any system The property has been shown to hold in a more general context, namely for fields rather than only for particle motion, by Noether We consider here only the ‘particle version’ of the theorem Before the embark on this theorem we will comment on what is meant by the statement that... Lagrangian Mechanics The results of variational calculus derived above allow us now to formulate the Hamiltonian Principle of Least Action of Classical Mechanics and study its equivalence to the Newtonian equations of motion Threorem: Hamiltonian Principle of Least Action The trajectories q(t) of systems of particles described through the Newtonian equations of motion d ∂U (mj qj ) + ˙ = 0 dt ∂qj ; j = . Hilbert space structure of Quantum Mechanics, on scattering theory, on perturbation theory, on Stochastic Quantum Mechanics, and on the group theory of elementary. Relativity and Quantum Mechanics. The notes are in a stage of continuing development, various sections, e.g., on the semiclassical approximation, on the Hilbert