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Eyal BuksQuantum Mechanics - LectureNotesDecember 23, 2012Technion
PrefaceThe dynamics of a quantum system is governed by the celebrated Schrödingerequation        d    i     |ψ = H |ψ ,   ...
Contents1.   Hamilton’s Formalism of Classical Physics . . . . . . . . . . . . . . . .                                    ...
Contents     3.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604...
Contents       9.4 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2...
Contents14. The Quantized Electromagnetic Field . . . . . . . . . . . . . . . . . . . . .                                 ...
Contents            17.3.3 Qubit Readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .            ...
1. Hamilton’s Formalism of Classical PhysicsIn this chapter the Hamilton’s formalism of classical physics is introduced,wi...
Chapter 1. Hamilton’s Formalism of Classical Physics                       Q              Q2              Q1              ...
1.2. Principle of Least Action     d ∂L      ∂L             =     ,                                                       ...
Chapter 1. Hamilton’s Formalism of Classical Physics                        Q               Q2                            ...
1.3. Hamiltonian1.3 HamiltonianThe set of Euler-Lagrange equations contains N second order differentialequations. In this s...
Chapter 1. Hamilton’s Formalism of Classical PhysicsThus the following holds            ∂H       qn =       ˙         ,   ...
1.4. Poisson’s Brackets          N    H=          pl ql − L                   ˙          l=1                ∂T      =     ...
Chapter 1. Hamilton’s Formalism of Classical Physics              N    dF              ∂F       ∂F       ∂F       =       ...
1.6. Solutions                            L               CFig. 1.3. LC resonator. 2. Consider an LC resonator made of a c...
Chapter 1. Hamilton’s Formalism of Classical Physics    a) The Euler-Lagrange equation for the coordinate x is given by   ...
1.6. Solutions            H = p·r−L                  ˙                          1      q               = r · p − m˙ − A + ...
Chapter 1. Hamilton’s Formalism of Classical Physics     c) Clearly, the fields E and B, which are given by Eqs. (1.41) and...
1.6. Solutions     c) Using the definition (1.37) one has                            ∂q ∂p ∂q ∂p               {q, p} =    ...
Chapter 1. Hamilton’s Formalism of Classical Physics          ∂ xb            ˙     ∂ 2 xb      ∂ 2 xb   d      ∂xb       ...
2. State Vectors and OperatorsIn quantum mechanics the state of a physical system is described by a statevector |α , which...
Chapter 2. State Vectors and Operators     β |α ∈ C ,                                                              (2.2)  ...
2.3. Dirac’s notation2.2 OperatorsOperators, as the definition below states, are function from F to F:Definition 2.2.1. An o...
Chapter 2. State Vectors and OperatorsWhile the multiplication of a bra-vector on the left and a ket-vector on theright re...
2.4. Dual Correspondence    F (c1 |γ 1 + c2 |γ 2 ) = c1 F (|γ 1 ) + c2 F (|γ 2 )                     (2.25)for every |γ 1 ...
Chapter 2. State Vectors and Operators                                      ∗    |β DD =                 β |φn         |φn...
2.5. Matrix Representation• The inner product between the bra-vector β| and the ket-vector |α can  be written as       β |...
Chapter 2. State Vectors and Operators• Such matrix representation of linear operators can be useful also for mul-  tiplyi...
2.6. Observables               †Claim. X †         =XProof. For any |α , |β ∈ F the following holds                       ...
Chapter 2. State Vectors and OperatorsClearly, Fn is closed under vector addition and scalar multiplication, namelyc1 |γ 1...
2.6. Observablesan orthonormal basis of the eigensubspace Fn , namely an,i′ |an,i = δ ii′ .Constructing such an orthonorma...
Chapter 2. State Vectors and Operatorscomplete. On the other hand, it can be shown that if a given Hermitian oper-ator A s...
2.6. Observableswhere           1 dm f    fm =          .                                                     (2.71)      ...
Chapter 2. State Vectors and OperatorsThus, the desired expression is given by               A − am    Pn =               ...
2.8. Example - Spin 1/2We also note that a direct consequence of the collapse postulate is that twosubsequent measurements...
Chapter 2. State Vectors and OperatorsThis result can be also written as              µ    µorbital = B L ,               ...
2.8. Example - Spin 1/2    |+ +| + |− −| = 1 .                                                 (2.97)In this basis any ket...
Quantum Mechanics: Lecture notes
Quantum Mechanics: Lecture notes
Quantum Mechanics: Lecture notes
Quantum Mechanics: Lecture notes
Quantum Mechanics: Lecture notes
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Quantum Mechanics: Lecture notes
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Quantum Mechanics: Lecture notes
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Quantum Mechanics: Lecture notes
Quantum Mechanics: Lecture notes
Quantum Mechanics: Lecture notes
Quantum Mechanics: Lecture notes
Quantum Mechanics: Lecture notes
Quantum Mechanics: Lecture notes
Quantum Mechanics: Lecture notes
Quantum Mechanics: Lecture notes
Quantum Mechanics: Lecture notes
Quantum Mechanics: Lecture notes
Quantum Mechanics: Lecture notes
Quantum Mechanics: Lecture notes
Quantum Mechanics: Lecture notes
Quantum Mechanics: Lecture notes
Quantum Mechanics: Lecture notes
Quantum Mechanics: Lecture notes
Quantum Mechanics: Lecture notes
Quantum Mechanics: Lecture notes
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Transcript of "Quantum Mechanics: Lecture notes"

  1. 1. Eyal BuksQuantum Mechanics - LectureNotesDecember 23, 2012Technion
  2. 2. PrefaceThe dynamics of a quantum system is governed by the celebrated Schrödingerequation d i |ψ = H |ψ , (0.1) dt √where i = −1 and = 1.05457266 × 10−34 J s is Planck’s h-bar constant.However, what is the meaning of the symbols |ψ and H? The answers willbe given in the first part of the course (chapters 1-4), which reviews severalphysical and mathematical concepts that are needed to formulate the theoryof quantum mechanics. We will learn that |ψ in Eq. (0.1) represents theket-vector state of the system and H represents the Hamiltonian operator.The operator H is directly related to the Hamiltonian function in classicalphysics, which will be defined in the first chapter. The ket-vector state andits physical meaning will be introduced in the second chapter. Chapter 3reviews the position and momentum operators, whereas chapter 4 discussesdynamics of quantum systems. The second part of the course (chapters 5-7)is devoted to some relatively simple quantum systems including a harmonicoscillator, spin, hydrogen atom and more. In chapter 8 we will study quantumsystems in thermal equilibrium. The third part of the course (chapters 9-13)is devoted to approximation methods such as perturbation theory, semiclas-sical and adiabatic approximations. Light and its interaction with matter arethe subjects of chapter 14-15. Finally, systems of identical particles will bediscussed in chapter 16 and open quantum systems in chapter 17. Most ofthe material in these lecture notes is based on the textbooks [1, 2, 3, 4, 6, 7].
  3. 3. Contents1. Hamilton’s Formalism of Classical Physics . . . . . . . . . . . . . . . . 1 1.1 Action and Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Principle of Least Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Poisson’s Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92. State Vectors and Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1 Linear Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3 Dirac’s notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 Dual Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.5 Matrix Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.6 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.6.1 Hermitian Adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.6.2 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . 23 2.7 Quantum Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.8 Example - Spin 1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.9 Unitary Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.10 Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.11 Commutation Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.12 Simultaneous Diagonalization of Commuting Operators . . . . . 35 2.13 Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.14 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.15 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403. The Position and Momentum Observables . . . . . . . . . . . . . . . . 49 3.1 The One Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.1.1 Position Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.1.2 Momentum Representation . . . . . . . . . . . . . . . . . . . . . . . . 54 3.2 Transformation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3 Generalization for 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
  4. 4. Contents 3.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604. Quantum Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.1 Time Evolution Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2 Time Independent Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.3 Example - Spin 1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.4 Connection to Classical Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 73 4.5 Symmetric Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795. The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.1 Eigenstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.2 Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.4 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106. Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.1 Angular Momentum and Rotation . . . . . . . . . . . . . . . . . . . . . . . . 138 6.2 General Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.3 Simultaneous Diagonalization of J2 and Jz . . . . . . . . . . . . . . . . 140 6.4 Example - Spin 1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.5 Orbital Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 6.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1587. Central Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 7.1 Simultaneous Diagonalization of the Operators H, L2 and Lz 186 7.2 The Radial Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 7.3 Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 7.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 7.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1978. Density Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 8.1 Time Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 8.2 Quantum Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 8.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 8.4 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2189. Time Independent Perturbation Theory . . . . . . . . . . . . . . . . . . 257 9.1 The Level En . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 9.1.1 Nondegenerate Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 9.1.2 Degenerate Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 9.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 9.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264Eyal Buks Quantum Mechanics - Lecture Notes 6
  5. 5. Contents 9.4 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26910. Time-Dependent Perturbation Theory . . . . . . . . . . . . . . . . . . . . 291 10.1 Time Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 10.2 Perturbation Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 10.3 The Operator O (t) = u† (t, t0 ) u (t, t0 ) . . . . . . . . . . . . . . . . . . . . . 0 293 10.4 Transition Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 10.4.1 The Stationary Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 10.4.2 The Near-Resonance Case . . . . . . . . . . . . . . . . . . . . . . . . . 297 10.4.3 H1 is Separable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 10.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 10.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30011. WKB Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 11.1 WKB Wavefunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 11.2 Turning Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 11.3 Bohr-Sommerfeld Quantization Rule . . . . . . . . . . . . . . . . . . . . . . 310 11.4 Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 11.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 11.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31312. Path Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 12.1 Charged Particle in Electromagnetic Field . . . . . . . . . . . . . . . . . 321 12.2 Classical Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 12.3 Aharonov-Bohm Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 12.3.1 Two-slit Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 12.3.2 Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 12.4 One Dimensional Path Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 331 12.4.1 One Dimensional Free Particle . . . . . . . . . . . . . . . . . . . . . 332 12.4.2 Expansion Around the Classical Path . . . . . . . . . . . . . . . 333 12.4.3 One Dimensional Harmonic Oscillator . . . . . . . . . . . . . . . 335 12.5 Semiclassical Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 12.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 12.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34113. Adiabatic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 13.1 Momentary Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 13.2 Gauge Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 13.3 Adiabatic Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 13.4 The Case of Two Dimensional Hilbert Space . . . . . . . . . . . . . . . 352 13.5 Transition Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 13.5.1 The Case of Two Dimensional Hilbert Space . . . . . . . . . 355 13.6 Slow and Fast Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 13.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 13.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361Eyal Buks Quantum Mechanics - Lecture Notes 7
  6. 6. Contents14. The Quantized Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . 363 14.1 Classical Electromagnetic Cavity . . . . . . . . . . . . . . . . . . . . . . . . . 363 14.2 Quantum Electromagnetic Cavity . . . . . . . . . . . . . . . . . . . . . . . . 368 14.3 Periodic Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 14.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 14.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37115. Light Matter Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 15.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 15.2 Transition Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 15.2.1 Spontaneous Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 15.2.2 Stimulated Emission and Absorption . . . . . . . . . . . . . . . . 381 15.2.3 Selection Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 15.3 Semiclassical Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 15.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 15.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38616. Identical Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 16.1 Basis for the Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 16.2 Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 16.3 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 16.4 Changing the Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 16.5 Many Particle Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 16.5.1 One-Particle Observables . . . . . . . . . . . . . . . . . . . . . . . . . . 399 16.5.2 Two-Particle Observables . . . . . . . . . . . . . . . . . . . . . . . . . . 400 16.6 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 16.7 Momentum Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 16.8 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 16.9 The Electron Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 16.10Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 16.11Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41017. Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 17.1 Macroscopic Wavefunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 17.1.1 Single Particle in Electromagnetic Field . . . . . . . . . . . . . 421 17.1.2 Drude Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 17.1.3 The Macroscopic Quantum Model . . . . . . . . . . . . . . . . . . 425 17.1.4 London Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 17.2 The Josephson Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 17.2.1 The First Josephson Relation . . . . . . . . . . . . . . . . . . . . . . 430 17.2.2 The Second Josephson Relation . . . . . . . . . . . . . . . . . . . . 431 17.2.3 The Energy of a Josephson Junction . . . . . . . . . . . . . . . . 432 17.3 RF SQUID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 17.3.1 Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 17.3.2 Flux Quantum Bit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436Eyal Buks Quantum Mechanics - Lecture Notes 8
  7. 7. Contents 17.3.3 Qubit Readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 17.4 BCS Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 17.4.1 Phonon Mediated Electron-Electron Interaction . . . . . . 445 17.4.2 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 17.4.3 Bogoliubov Transformation . . . . . . . . . . . . . . . . . . . . . . . . 449 17.4.4 The Energy Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 17.4.5 The Ground State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 17.4.6 Pairing Wavefunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 17.5 The Josephson Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 17.5.1 The Second Josephson Relation . . . . . . . . . . . . . . . . . . . . 459 17.5.2 The Energy of a Josephson Junction . . . . . . . . . . . . . . . . 460 17.5.3 The First Josephson Relation . . . . . . . . . . . . . . . . . . . . . . 463 17.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 17.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46518. Open Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 18.1 Classical Resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 18.2 Quantum Resonator Coupled to Thermal Bath . . . . . . . . . . . . . 470 18.2.1 The closed System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 18.2.2 Coupling to Thermal Bath . . . . . . . . . . . . . . . . . . . . . . . . 471 18.2.3 Thermal Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 18.3 Two Level System Coupled to Thermal Bath . . . . . . . . . . . . . . . 477 18.3.1 The Closed System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 18.3.2 Coupling to Thermal Baths . . . . . . . . . . . . . . . . . . . . . . . . 478 18.3.3 Thermal Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 18.3.4 Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 18.3.5 The Bloch Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 18.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 18.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495Eyal Buks Quantum Mechanics - Lecture Notes 9
  8. 8. 1. Hamilton’s Formalism of Classical PhysicsIn this chapter the Hamilton’s formalism of classical physics is introduced,with a special emphasis on the concepts that are needed for quantum me-chanics.1.1 Action and LagrangianConsider a classical physical system having N degrees of freedom. The clas-sical state of the system can be described by N independent coordinates qn ,where n = 1, 2, · · · , N . The vector of coordinates is denoted by Q = (q1 , q2 , · · · , qN ) . (1.1)Consider the case where the vector of coordinates takes the value Q1 at timet1 and the value Q2 at a later time t2 > t1 , namely Q (t1 ) = Q1 , (1.2) Q (t2 ) = Q2 . (1.3)The action S associated with the evolution of the system from time t1 totime t2 is defined by t2 S= dt L , (1.4) t1where L is the Lagrangian function of the system. In general, the Lagrangian ˙is a function of the coordinates Q, the velocities Q and time t, namely ˙ L = L Q, Q; t , (1.5)where ˙ Q = (q1 , q2 , · · · , qN ) , ˙ ˙ ˙ (1.6)and where overdot denotes time derivative. The time evolution of Q, in turn,depends of the trajectory taken by the system from point Q1 at time t1
  9. 9. Chapter 1. Hamilton’s Formalism of Classical Physics Q Q2 Q1 t t1 t2Fig. 1.1. A trajectory taken by the system from point Q1 at time t1 to point Q2at time t2 .to point Q2 at time t2 (see Fig. 1.1). For a given trajectory Γ the timedependency is denoted as Q (t) = QΓ (t) . (1.7)1.2 Principle of Least ActionFor any given trajectory Q (t) the action can be evaluated using Eq. (1.4).Consider a classical system evolving in time from point Q1 at time t1 to pointQ2 at time t2 along the trajectory QΓ (t). The trajectory QΓ (t), which isobtained from the laws of classical physics, has the following unique propertyknown as the principle of least action:Proposition 1.2.1 (principle of least action). Among all possible trajec-tories from point Q1 at time t1 to point Q2 at time t2 the action obtains itsminimal value by the classical trajectory QΓ (t). In a weaker version of this principle, the action obtains a local minimumfor the trajectory QΓ (t). As the following theorem shows, the principle ofleast action leads to a set of equations of motion, known as Euler-Lagrangeequations.Theorem 1.2.1. The classical trajectory QΓ (t), for which the action obtainsits minimum value, obeys the Euler-Lagrange equations of motion, which aregiven byEyal Buks Quantum Mechanics - Lecture Notes 2
  10. 10. 1.2. Principle of Least Action d ∂L ∂L = , (1.8) dt ∂ qn ˙ ∂qnwhere n = 1, 2, · · · , N.Proof. Consider another trajectory QΓ ′ (t) from point Q1 at time t1 to pointQ2 at time t2 (see Fig. 1.2). The difference δQ = QΓ ′ (t) − QΓ (t) = (δq1 , δq2 , · · · , δqN ) (1.9)is assumed to be infinitesimally small. To lowest order in δQ the change inthe action δS is given by t2 δS = dt δL t1 t2 N N ∂L ∂L = dt δqn + δ qn ˙ n=1 ∂qn n=1 ∂ qn ˙ t1 t2 N N ∂L ∂L d = dt δqn + δqn . n=1 ∂qn n=1 ∂ qn dt ˙ t1 (1.10)Integrating the second term by parts leads to t2 N ∂L d ∂L δS = dt − δqn n=1 ∂qn dt ∂ qn ˙ t1 N t2 ∂L + δqn . n=1 ∂ qn ˙ t1 (1.11)The last term vanishes since δQ (t1 ) = δQ (t2 ) = 0 . (1.12)The principle of least action implies that δS = 0 . (1.13)This has to be satisfied for any δQ, therefore the following must hold d ∂L ∂L = . (1.14) dt ∂ qn ˙ ∂qnEyal Buks Quantum Mechanics - Lecture Notes 3
  11. 11. Chapter 1. Hamilton’s Formalism of Classical Physics Q Q2 Γ Γ’ Q1 t t1 t2Fig. 1.2. The classical trajectory QΓ (t) and the trajectory QΓ ′ (t). In what follows we will assume for simplicity that the kinetic energy T of ˙the system can be expressed as a function of the velocities Q only (namely,it does not explicitly depend on the coordinates Q). The components of thegeneralized force Fn , where n = 1, 2, · · · , N , are derived from the potentialenergy U of the system as follows ∂U d ∂U Fn = − + . (1.15) ∂qn dt ∂ qn ˙When the potential energy can be expressed as a function of the coordinates ˙Q only (namely, when it is independent on the velocities Q), the system issaid to be conservative. For that case, the Lagrangian can be expressed interms of T and U as L=T −U . (1.16)Example 1.2.1. Consider a point particle having mass m moving in a one-dimensional potential U (x). The Lagrangian is given by mx2 ˙ L=T −U = − U (x) . (1.17) 2From the Euler-Lagrange equation d ∂L ∂L = , (1.18) dt ∂ x ˙ ∂xone finds that ∂U m¨ = − x . (1.19) ∂xEyal Buks Quantum Mechanics - Lecture Notes 4
  12. 12. 1.3. Hamiltonian1.3 HamiltonianThe set of Euler-Lagrange equations contains N second order differentialequations. In this section we derive an alternative and equivalent set of equa-tions of motion, known as Hamilton-Jacobi equations, that contains twice thenumber of equations, namely 2N, however, of first, instead of second, order.Definition 1.3.1. The variable canonically conjugate to qn is defined by ∂L pn = . (1.20) ∂ qn ˙Definition 1.3.2. The Hamiltonian of a physical system is a function ofthe vector of coordinates Q, the vector of canonical conjugate variables P =(p1 , p2 , · · · , pN ) and time, namely H = H (Q, P ; t) , (1.21)is defined by N H= pn qn − L , ˙ (1.22) n=1where L is the Lagrangian.Theorem 1.3.1. The classical trajectory satisfies the Hamilton-Jacobi equa-tions of motion, which are given by ∂H qn = ˙ , (1.23) ∂pn ∂H pn = − ˙ , (1.24) ∂qnwhere n = 1, 2, · · · , N.Proof. The differential of H is given by N dH = d pn qn − dL ˙ n=1   N    = qn dpn + pn dqn − ∂L dqn − ∂L dqn  − ∂L dt ˙ ˙ ∂qn ∂ qn ˙ ˙  n=1   ∂t d ∂L pn dt ∂ qn ˙ N ∂L = (qn dpn − pn dqn ) − ˙ ˙ dt . n=1 ∂t (1.25)Eyal Buks Quantum Mechanics - Lecture Notes 5
  13. 13. Chapter 1. Hamilton’s Formalism of Classical PhysicsThus the following holds ∂H qn = ˙ , (1.26) ∂pn ∂H pn = − ˙ , (1.27) ∂qn ∂L ∂H − = . (1.28) ∂t ∂tCorollary 1.3.1. The following holds dH ∂H = . (1.29) dt ∂tProof. Using Eqs. (1.23) and (1.24) one finds that N dH ∂H ∂H ∂H ∂H = qn + ˙ pn + ˙ = . (1.30) dt n=1 ∂qn ∂pn ∂t ∂t =0 The last corollary implies that H is time independent provided that Hdoes not depend on time explicitly, namely, provided that ∂H/∂t = 0. Thisproperty is referred to as the law of energy conservation. The theorem belowfurther emphasizes the relation between the Hamiltonian and the total energyof the system.Theorem 1.3.2. Assume that the kinetic energy of a conservative system isgiven by T = αnm qn qm , ˙ ˙ (1.31) n,mwhere αnm are constants. Then,the Hamiltonian of the system is given by H =T +U , (1.32)where T is the kinetic energy of the system and where U is the potentialenergy.Proof. For a conservative system the potential energy is independent on ve-locities, thus ∂L ∂T pl = = , (1.33) ∂ ql ˙ ∂ ql ˙where L = T − U is the Lagrangian. The Hamiltonian is thus given byEyal Buks Quantum Mechanics - Lecture Notes 6
  14. 14. 1.4. Poisson’s Brackets N H= pl ql − L ˙ l=1 ∂T = ql − (T − U ) ˙ ∂ ql ˙ l    ∂q˙n ∂ qm  ˙   = αnm qm ˙ + qn ˙  ql − T + U ˙  ∂ ql ˙ ∂ ql  ˙ l,n,m δnl δ ml =2 αnm qn qm − T + U ˙ ˙ n,m T = T +U . (1.34)1.4 Poisson’s BracketsConsider two physical quantities F and G that can be expressed as a functionof the vector of coordinates Q, the vector of canonical conjugate variables Pand time t, namely F = F (Q, P ; t) , (1.35) G = G (Q, P ; t) , (1.36)The Poisson’s brackets are defined by N ∂F ∂G ∂F ∂G {F, G} = − , (1.37) n=1 ∂qn ∂pn ∂pn ∂qnThe Poisson’s brackets are employed for writing an equation of motion for ageneral physical quantity of interest, as the following theorem shows.Theorem 1.4.1. Let F be a physical quantity that can be expressed as afunction of the vector of coordinates Q, the vector of canonical conjugatevariables P and time t, and let H be the Hamiltonian. Then, the followingholds dF ∂F = {F, H} + . (1.38) dt ∂tProof. Using Eqs. (1.23) and (1.24) one finds that the time derivative of Fis given byEyal Buks Quantum Mechanics - Lecture Notes 7
  15. 15. Chapter 1. Hamilton’s Formalism of Classical Physics N dF ∂F ∂F ∂F = qn + ˙ pn + ˙ dt n=1 ∂qn ∂pn ∂t N ∂F ∂H ∂F ∂H ∂F = − + n=1 ∂qn ∂pn ∂pn ∂qn ∂t ∂F = {F, H} + . ∂t (1.39)Corollary 1.4.1. If F does not explicitly depend on time, namely if ∂F/∂t =0, and if {F, H} = 0, then F is a constant of the motion, namely dF =0. (1.40) dt1.5 Problems 1. Consider a particle having charge q and mass m in electromagnetic field characterized by the scalar potential ϕ and the vector potential A. The electric field E and the magnetic field B are given by 1 ∂A E = −∇ϕ − , (1.41) c ∂t and B=∇×A. (1.42) Let r = (x, y, z) be the Cartesian coordinates of the particle. a) Verify that the Lagrangian of the system can be chosen to be given by 1 2 q L= m˙ − qϕ + A · r , r ˙ (1.43) 2 c by showing that the corresponding Euler-Lagrange equations are equivalent to Newton’s 2nd law (i.e., F = m¨). r b) Show that the Hamilton-Jacobi equations are equivalent to Newton’s 2nd law. c) Gauge transformation — The electromagnetic field is invariant un- der the gauge transformation of the scalar and vector potentials A → A + ∇χ , (1.44) 1 ∂χ ϕ → ϕ− (1.45) c ∂t where χ = χ (r, t) is an arbitrary smooth and continuous function of r and t. What effect does this gauge transformation have on the Lagrangian and Hamiltonian? Is the motion affected?Eyal Buks Quantum Mechanics - Lecture Notes 8
  16. 16. 1.6. Solutions L CFig. 1.3. LC resonator. 2. Consider an LC resonator made of a capacitor having capacitance C in parallel with an inductor having inductance L (see Fig. 1.3). The state of the system is characterized by the coordinate q , which is the charge stored by the capacitor. a) Find the Euler-Lagrange equation of the system. b) Find the Hamilton-Jacobi equations of the system. c) Show that {q, p} = 1 . 3. Show that Poisson brackets satisfy the following relations {qj , qk } = 0 , (1.46) {pj , pk } = 0 , (1.47) {qj , pk } = δ jk , (1.48) {F, G} = − {G, F } , (1.49) {F, F } = 0 , (1.50) {F, K} = 0 if K constant or F depends only on t , (1.51) {E + F, G} = {E, G} + {F, G} , (1.52) {E, F G} = {E, F } G + F {E, G} . (1.53) 4. Show that the Lagrange equations are coordinate invariant. 5. Consider a point particle having mass m moving in a 3D central po- tential, namely a potential V (r) that depends only on the distance r = x2 + y 2 + z 2 from the origin. Show that the angular momentum L = r × p is a constant of the motion.1.6 Solutions 1. The Lagrangian of the system (in Gaussian units) is taken to be given by 1 q L = m˙ 2 − qϕ + A · r . r ˙ (1.54) 2 cEyal Buks Quantum Mechanics - Lecture Notes 9
  17. 17. Chapter 1. Hamilton’s Formalism of Classical Physics a) The Euler-Lagrange equation for the coordinate x is given by d ∂L ∂L = , (1.55) dt ∂ x ˙ ∂x where d ∂L q ∂Ax ∂Ax ∂Ax ∂Ax = m¨ + x +x ˙ +y ˙ +z ˙ , (1.56) dt ∂ x ˙ c ∂t ∂x ∂y ∂z and ∂L ∂ϕ q ∂Ax ∂Ay ∂Az = −q + x ˙ +y ˙ +z ˙ , (1.57) ∂x ∂x c ∂x ∂x ∂x thus ∂ϕ q ∂Ax m¨ = −q x − ∂x c ∂t qEx           q  ∂Ay ∂Ax ∂Ax ∂Az  + y ˙ − −z ˙ −  , c ∂x ∂y ∂z ∂x      (∇ ×A)z (∇ ×A)y    (˙ ×(∇×A))x r (1.58) or q m¨ = qEx + x (˙ × B)x . r (1.59) c Similar equations are obtained for y and z in the same way. These 3 ¨ ¨ equations can be written in a vector form as 1 m¨ = q E + r × B r ˙ . (1.60) c b) The variable vector canonically conjugate to the coordinates vector r is given by ∂L q p= = m˙ + A . r (1.61) ∂r ˙ c The Hamiltonian is thus given byEyal Buks Quantum Mechanics - Lecture Notes 10
  18. 18. 1.6. Solutions H = p·r−L ˙ 1 q = r · p − m˙ − A + qϕ ˙ r 2 c 1 2 = m˙ + qϕ r 2 2 p− q A c = + qϕ . 2m (1.62) The Hamilton-Jacobi equation for the coordinate x is given by ∂H x= ˙ , (1.63) ∂px thus px − q Ax c x= ˙ , (1.64) m or q px = mx+ Ax . ˙ (1.65) c The Hamilton-Jacobi equation for the canonically conjugate variable px is given by ∂H px = − ˙ , (1.66) ∂x where q ∂Ax ∂Ax ∂Ax q ∂Ax px = m¨+ ˙ x x ˙ +y ˙ +z ˙ + , (1.67) c ∂x ∂y ∂z c ∂t and ∂H q px − q Ax ∂Ax py − q Ay ∂Ay c c pz − q Az ∂Az c ∂ϕ − = + + −q ∂x c m ∂x m ∂x m ∂x ∂x q ∂Ax ∂Ay ∂Az ∂ϕ = x ˙ +y ˙ +z ˙ −q , c ∂x ∂x ∂x ∂x (1.68) thus ∂ϕ q ∂Ax q ∂Ay ∂Ax ∂Ax ∂Az m¨ = −q x − + y ˙ − −z ˙ − . ∂x c ∂t c ∂x ∂y ∂z ∂x (1.69) The last result is identical to Eq. (1.59).Eyal Buks Quantum Mechanics - Lecture Notes 11
  19. 19. Chapter 1. Hamilton’s Formalism of Classical Physics c) Clearly, the fields E and B, which are given by Eqs. (1.41) and (1.42) respectively, are unchanged since ∂χ ∂ (∇χ) ∇ − =0, (1.70) ∂t ∂t and ∇ × (∇χ) = 0 . (1.71) Thus, even though both L and H are modified, the motion, which depends on E and B only, is unaffected. 2. The kinetic energy in this case T = Lq 2 /2 is the energy stored in the ˙ inductor, and the potential energy U = q 2 /2C is the energy stored in the capacitor. a) The Lagrangian is given by Lq 2 ˙ q2 L=T −U = − . (1.72) 2 2C The Euler-Lagrange equation for the coordinate q is given by d ∂L ∂L = , (1.73) dt ∂ q ˙ ∂q thus q L¨ + q =0. (1.74) C This equation expresses the requirement that the voltage across the capacitor is the same as the one across the inductor. b) The canonical conjugate momentum is given by ∂L p= = Lq , ˙ (1.75) ∂q ˙ and the Hamiltonian is given by p2 q2 H = pq − L = ˙ + . (1.76) 2L 2C Hamilton-Jacobi equations read p q= ˙ (1.77) L q p=− , ˙ (1.78) C thus q L¨ + q =0. (1.79) CEyal Buks Quantum Mechanics - Lecture Notes 12
  20. 20. 1.6. Solutions c) Using the definition (1.37) one has ∂q ∂p ∂q ∂p {q, p} = − =1. (1.80) ∂q ∂p ∂p ∂q 3. All these relations are easily proven using the definition (1.37). ˙ 4. Let L = L Q, Q; t be a Lagrangian of a system, where Q = (q1 , q2 , · · · ) ˙ is the vector of coordinates, Q = (q1 , q2 , · · · ) is the vector of veloci- ˙ ˙ ties, and where overdot denotes time derivative. Consider the coordinates transformation xa = xa (q1 , q2 , ..., t) , (1.81) where a = 1, 2, · · · . The following holds ∂xa ∂xa xa = ˙ qb + ˙ , (1.82) ∂qb ∂t where the summation convention is being used, namely, repeated indices are summed over. Moreover ∂L ∂L ∂xb ∂L ∂ xb ˙ = + , (1.83) ∂qa ∂xb ∂qa ∂ xb ∂qa ˙ and d ∂L d ∂L ∂ xb˙ = . (1.84) dt ∂ qa ˙ dt ∂ xb ∂ qa ˙ ˙ As can be seen from Eq. (1.82), one has ∂ xb ˙ ∂xb = . (1.85) ∂ qa ˙ ∂qa Thus, using Eqs. (1.83) and (1.84) one finds d ∂L ∂L d ∂L ∂xb − = dt ∂ qa ˙ ∂qa dt ∂ xb ∂qa ˙ ∂L ∂xb ∂L ∂ xb˙ − − ∂xb ∂qa ∂ xb ∂qa ˙ d ∂L ∂L ∂xb = − dt ∂ xb˙ ∂xb ∂qa d ∂xb ∂ xb ˙ ∂L + − . dt ∂qa ∂qa ∂ xb ˙ (1.86) As can be seen from Eq. (1.82), the second term vanishes sinceEyal Buks Quantum Mechanics - Lecture Notes 13
  21. 21. Chapter 1. Hamilton’s Formalism of Classical Physics ∂ xb ˙ ∂ 2 xb ∂ 2 xb d ∂xb = qc + ˙ = , ∂qa ∂qa ∂qc ∂t∂qa dt ∂qa thus d ∂L ∂L d ∂L ∂L ∂xb − = − . (1.87) dt ∂ qa ˙ ∂qa dt ∂ xb ˙ ∂xb ∂qa The last result shows that if the coordinate transformation is reversible, namely if det (∂xb /∂qa ) = 0 then Lagrange equations are coordinate invariant. 5. The angular momentum L is given by   x y ˆ ˆ ˆ z L = r × p = det  x y z  , (1.88) px py pz where r = (x, y, z) is the position vector and where p = (px , py , pz ) is the momentum vector. The Hamiltonian is given by p2 H= + V (r) . (1.89) 2m Using {xi , pj } = δ ij , (1.90) Lz = xpy − ypx , (1.91) one finds that p2 , Lz = p2 , Lz + p2 , Lz + p2 , Lz x y z = p2 , xpy − p2 , ypx x y = −2px py + 2py px =0, (1.92) and r2 , Lz = x2 , Lz + y 2 , Lz + z 2 , Lz = −y x2 , px + y 2 , py x =0. (1.93) Thus f r2 , Lz = 0 for arbitrary smooth function f r2 , and con- sequently {H, Lz } = 0. In a similar way one can show that {H, Lx } = {H, Ly } = 0, and therefore H, L2 = 0.Eyal Buks Quantum Mechanics - Lecture Notes 14
  22. 22. 2. State Vectors and OperatorsIn quantum mechanics the state of a physical system is described by a statevector |α , which is a vector in a vector space F, namely |α ∈ F . (2.1)Here, we have employed the Dirac’s ket-vector notation |α for the state vec-tor, which contains all information about the state of the physical systemunder study. The dimensionality of F is finite in some specific cases (no-tably, spin systems), however, it can also be infinite in many other casesof interest. The basic mathematical theory dealing with vector spaces hav-ing infinite dimensionality was mainly developed by David Hilbert. Undersome conditions, vector spaces having infinite dimensionality have propertiessimilar to those of their finite dimensionality counterparts. A mathematicallyrigorous treatment of such vector spaces having infinite dimensionality, whichare commonly called Hilbert spaces, can be found in textbooks that are de-voted to this subject. In this chapter, however, we will only review the mainproperties that are useful for quantum mechanics. In some cases, when thegeneralization from the case of finite dimensionality to the case of arbitrarydimensionality is nontrivial, results will be presented without providing arigorous proof and even without accurately specifying what are the validityconditions for these results.2.1 Linear Vector SpaceA linear vector space F is a set {|α } of mathematical objects called vectors.The space is assumed to be closed under vector addition and scalar multipli-cation. Both, operations (i.e., vector addition and scalar multiplication) arecommutative. That is: 1. |α + |β = |β + |α ∈ F for every |α ∈ F and |β ∈ F 2. c |α = |α c ∈ F for every |α ∈ F and c ∈ C (where C is the set of complex numbers) A vector space with an inner product is called an inner product space.An inner product of the ordered pair |α , |β ∈ F is denoted as β |α . Theinner product is a function F 2 → C that satisfies the following properties:
  23. 23. Chapter 2. State Vectors and Operators β |α ∈ C , (2.2) ∗ β |α = α |β , (2.3) α (c1 |β 1 + c2 |β 2 ) = c1 α |β 1 + c2 α |β 2 , where c1 , c2 ∈ C , (2.4) α |α ∈ R and α |α ≥ 0. Equality holds iff |α = 0 . (2.5)Note that the asterisk in Eq. (2.3) denotes complex conjugate. Below we listsome important definitions and comments regarding inner product:• The real number α |α is called the norm of the vector |α ∈ F.• A normalized vector has a unity norm, namely α |α = 1.• Every nonzero vector 0 = |α ∈ F can be normalized using the transfor- mation |α |α → . (2.6) α |α• The vectors |α ∈ F and |β ∈ F are said to be orthogonal if β |α = 0.• A set of vectors {|φn }n , where |φn ∈ F is called a complete orthonormal basis if — The vectors are all normalized and orthogonal to each other, namely φm |φn = δ nm . (2.7) — Every |α ∈ F can be written as a superposition of the basis vectors, namely |α = cn |φn , (2.8) n where cn ∈ C.• By evaluating the inner product φm |α , where |α is given by Eq. (2.8) one finds with the help of Eq. (2.7) and property (2.4) of inner products that φm |α = φm cn |φn = cn φm |φn = cm . (2.9) n n =δnm• The last result allows rewriting Eq. (2.8) as |α = cn |φn = |φn cn = |φn φn |α . (2.10) n n nEyal Buks Quantum Mechanics - Lecture Notes 16
  24. 24. 2.3. Dirac’s notation2.2 OperatorsOperators, as the definition below states, are function from F to F:Definition 2.2.1. An operator A : F → F on a vector space maps vectorsonto vectors, namely A |α ∈ F for every |α ∈ F. Some important definitions and comments are listed below:• The operators X : F → F and Y : F → F are said to be equal, namely X = Y , if for every |α ∈ F the following holds X |α = Y |α . (2.11)• Operators can be added, and the addition is both, commutative and asso- ciative, namely X +Y = Y +X , (2.12) X + (Y + Z) = (X + Y ) + Z . (2.13)• An operator A : F → F is said to be linear if A (c1 |γ 1 + c2 |γ 2 ) = c1 A |γ 1 + c2 A |γ 2 (2.14) for every |γ 1 , |γ 2 ∈ F and c1 , c2 ∈ C.• The operators X : F → F and Y : F → F can be multiplied, where XY |α = X (Y |α ) (2.15) for any |α ∈ F.• Operator multiplication is associative X (Y Z) = (XY ) Z = XY Z . (2.16)• However, in general operator multiplication needs not be commutative XY = Y X . (2.17)2.3 Dirac’s notationIn Dirac’s notation the inner product is considered as a multiplication of twomathematical objects called ’bra’ and ’ket’ β |α = β| |α . (2.18) bra ketWhile the ket-vector |α is a vector in F, the bra-vector β| represents afunctional that maps any ket-vector |α ∈ F to the complex number β |α .Eyal Buks Quantum Mechanics - Lecture Notes 17
  25. 25. Chapter 2. State Vectors and OperatorsWhile the multiplication of a bra-vector on the left and a ket-vector on theright represents inner product, the outer product is obtained by reversing theorder Aαβ = |α β| . (2.19)The outer product Aαβ is clearly an operator since for any |γ ∈ F the objectAαβ |γ is a ket-vector Aαβ |γ = (|β α|) |γ = |β α |γ ∈ F . (2.20) ∈CMoreover, according to property (2.4), Aαβ is linear since for every |γ 1 , |γ 2 ∈F and c1 , c2 ∈ C the following holds Aαβ (c1 |γ 1 + c2 |γ 2 ) = |α β| (c1 |γ 1 + c2 |γ 2 ) = |α (c1 β |γ 1 + c2 β |γ 2 ) = c1 Aαβ |γ 1 + c2 Aαβ |γ 2 . (2.21) With Dirac’s notation Eq. (2.10) can be rewritten as |α = |φn φn | |α . (2.22) nSince the above identity holds for any |α ∈ F one concludes that the quantityin brackets is the identity operator, which is denoted as 1, namely 1= |φn φn | . (2.23) nThis result, which is called the closure relation, implies that any completeorthonormal basis can be used to express the identity operator.2.4 Dual CorrespondenceAs we have mentioned above, the bra-vector β| represents a functional map-ping any ket-vector |α ∈ F to the complex number β |α . Moreover, sincethe inner product is linear [see property (2.4) above], such a mapping is linear,namely for every |γ 1 , |γ 2 ∈ F and c1 , c2 ∈ C the following holds β| (c1 |γ 1 + c2 |γ 2 ) = c1 β |γ 1 + c2 β |γ 2 . (2.24)The set of linear functionals from F to C, namely, the set of functionals F : F→ C that satisfyEyal Buks Quantum Mechanics - Lecture Notes 18
  26. 26. 2.4. Dual Correspondence F (c1 |γ 1 + c2 |γ 2 ) = c1 F (|γ 1 ) + c2 F (|γ 2 ) (2.25)for every |γ 1 , |γ 2 ∈ F and c1 , c2 ∈ C, is called the dual space F ∗ . Asthe name suggests, there is a dual correspondence (DC) between F and F ∗ ,namely a one to one mapping between these two sets, which are both linearvector spaces. The duality relation is presented using the notation α| ⇔ |α , (2.26)where |α ∈ F and α| ∈ F ∗ . What is the dual of the ket-vector |γ =c1 |γ 1 + c2 |γ 2 , where |γ 1 , |γ 2 ∈ F and c1 , c2 ∈ C? To answer this questionwe employ the above mentioned general properties (2.3) and (2.4) of innerproducts and consider the quantity γ |α for an arbitrary ket-vector |α ∈ F γ |α = α |γ ∗ = (c1 α |γ 1 + c2 α |γ 2 )∗ = c∗ γ 1 |α + c2 γ 2 |α 1 = (c∗ γ 1 | + c∗ γ 2 |) |α . 1 2 (2.27)From this result we conclude that the duality relation takes the form c∗ γ 1 | + c∗ γ 2 | ⇔ c1 |γ 1 + c2 |γ 2 . 1 2 (2.28) The last relation describes how to map any given ket-vector |β ∈ Fto its dual F = β| : F → C, where F ∈ F ∗ is a linear functional thatmaps any ket-vector |α ∈ F to the complex number β |α . What is theinverse mapping? The answer can take a relatively simple form provided thata complete orthonormal basis exists, and consequently the identity operatorcan be expressed as in Eq. (2.23). In that case the dual of a given linearfunctional F : F → C is the ket-vector |FD ∈ F, which is given by |FD = (F (|φn ))∗ |φn . (2.29) nThe duality is demonstrated by proving the two claims below:Claim. |β DD = |β for any |β ∈ F, where |β DD is the dual of the dual of|β .Proof. The dual of |β is the bra-vector β|, whereas the dual of β| is foundusing Eqs. (2.29) and (2.23), thusEyal Buks Quantum Mechanics - Lecture Notes 19
  27. 27. Chapter 2. State Vectors and Operators ∗ |β DD = β |φn |φn n = φn |β = |φn φn |β n = |φn φn | |β n =1 = |β . (2.30)Claim. FDD = F for any F ∈ F ∗ , where FDD is the dual of the dual of F .Proof. The dual |FD ∈ F of the functional F ∈ F ∗ is given by Eq. (2.29).Thus with the help of the duality relation (2.28) one finds that dual FDD ∈ F ∗of |FD is given by FDD = F (|φn ) φn | . (2.31) nConsider an arbitrary ket-vector |α ∈ F that is written as a superpositionof the complete orthonormal basis vectors, namely |α = cm |φm . (2.32) mUsing the above expression for FDD and the linearity property one finds that FDD |α = cm F (|φn ) φn |φm n,m δmn = cn F (|φn ) n =F cn |φn n = F (|α ) , (2.33)therefore, FDD = F .2.5 Matrix RepresentationGiven a complete orthonormal basis, ket-vectors, bra-vectors and linear op-erators can be represented using matrices. Such representations are easilyobtained using the closure relation (2.23).Eyal Buks Quantum Mechanics - Lecture Notes 20
  28. 28. 2.5. Matrix Representation• The inner product between the bra-vector β| and the ket-vector |α can be written as β |α = β| 1 |α = β |φn φn | α n   φ1 |α   = β |φ1 β |φ2 · · ·  φ2 |α  . . . . (2.34) Thus, the inner product can be viewed as a product between the row vector β| = β |φ1 ˙ β |φ2 · · · , (2.35) which is the matrix representation of the bra-vector β|, and the column vector   φ1 |α   |α =  φ2 |α  , ˙ (2.36) . . . which is the matrix representation of the ket-vector |α . Obviously, both representations are basis dependent.• Multiplying the relation |γ = X |α from the right by the basis bra-vector φm | and employing again the closure relation (2.23) yields φm |γ = φm | X |α = φm | X1 |α = φm | X |φn φn |α , (2.37) n or in matrix form      φ1 |γ φ1 | X |φ1 φ1 | X |φ2 · · · φ1 |α  φ2 |γ   φ2 | X |φ1 φ2 | X |φ2 · · ·  φ2 |α  .  =   (2.38) . . . . . . . . . . . . In view of this expression, the matrix representation of the linear operator X is given by   φ1 | X |φ1 φ1 | X |φ2 · · ·   X =  φ2 | X |φ1 φ2 | X |φ2 · · · . ˙ (2.39) . . . . . . Alternatively, the last result can be written as Xnm = φn | X |φm , (2.40) where Xnm is the element in row n and column m of the matrix represen- tation of the operator X.Eyal Buks Quantum Mechanics - Lecture Notes 21
  29. 29. Chapter 2. State Vectors and Operators• Such matrix representation of linear operators can be useful also for mul- tiplying linear operators. The matrix elements of the product Z = XY are given by φm | Z |φn = φm | XY |φn = φm | X1Y |φn = φm | X |φl φl | Y |φn . l (2.41)• Similarly, the matrix representation of the outer product |β α| is given by   φ1 |β   |β α| =  φ2 |β  α |φ1 α |φ2 · · · ˙ . . .   φ1 |β α |φ1 φ1 |β α |φ2 · · ·   =  φ2 |β α |φ1 φ2 |β α |φ2 · · · . . . . . . . (2.42)2.6 ObservablesMeasurable physical variables are represented in quantum mechanics by Her-mitian operators.2.6.1 Hermitian AdjointDefinition 2.6.1. The Hermitian adjoint of an operator X is denoted as X †and is defined by the following duality relation α| X † ⇔ X |α . (2.43)Namely, for any ket-vector |α ∈ F, the dual to the ket-vector X |α is thebra-vector α| X † .Definition 2.6.2. An operator is said to be Hermitian if X = X † . Below we prove some simple relations: ∗Claim. β| X |α = α| X † |βProof. Using the general property (2.3) of inner products one has ∗ ∗ β| X |α = β| (X |α ) = α| X † |β = α| X † |β . (2.44)Note that this result implies that if X = X † then β| X |α = α| X |β ∗ .Eyal Buks Quantum Mechanics - Lecture Notes 22
  30. 30. 2.6. Observables †Claim. X † =XProof. For any |α , |β ∈ F the following holds ∗ ∗ ∗ † β| X |α = β| X |α = α| X † |β = β| X † |α , (2.45) †thus X † = X.Claim. (XY )† = Y † X †Proof. Applying XY on an arbitrary ket-vector |α ∈ F and employing theduality correspondence yield XY |α = X (Y |α ) ⇔ α| Y † X † = α| Y † X † , (2.46)thus (XY )† = Y † X † . (2.47)Claim. If X = |β α| then X † = |α β|Proof. By applying X on an arbitrary ket-vector |γ ∈ F and employing theduality correspondence one finds that X |γ = (|β α|) |γ = |β ( α |γ ) ⇔ ( α |γ )∗ β| = γ |α β| = γ| X † , (2.48)where X † = |α β|.2.6.2 Eigenvalues and EigenvectorsEach operator is characterized by its set of eigenvalues, which is definedbelow:Definition 2.6.3. A number an ∈ C is said to be an eigenvalue of an op-erator A : F → F if for some nonzero ket-vector |an ∈ F the followingholds A |an = an |an . (2.49)The ket-vector |an is then said to be an eigenvector of the operator A withan eigenvalue an . The set of eigenvectors associated with a given eigenvalue of an operatorA is called eigensubspace and is denoted as Fn = {|an ∈ F such that A |an = an |an } . (2.50)Eyal Buks Quantum Mechanics - Lecture Notes 23
  31. 31. Chapter 2. State Vectors and OperatorsClearly, Fn is closed under vector addition and scalar multiplication, namelyc1 |γ 1 + c2 |γ 2 ∈ Fn for every |γ 1 , |γ 2 ∈ Fn and for every c1 , c2 ∈ C. Thus,the set Fn is a subspace of F. The dimensionality of Fn (i.e., the minimumnumber of vectors that are needed to span Fn ) is called the level of degeneracygn of the eigenvalue an , namely gn = dim Fn . (2.51) As the theorem below shows, the eigenvalues and eigenvectors of a Her-mitian operator have some unique properties.Theorem 2.6.1. The eigenvalues of a Hermitian operator A are real. Theeigenvectors of A corresponding to different eigenvalues are orthogonal.Proof. Let a1 and a2 be two eigenvalues of A with corresponding eigen vectors|a1 and |a2 A |a1 = a1 |a1 , (2.52) A |a2 = a2 |a2 . (2.53)Multiplying Eq. (2.52) from the left by the bra-vector a2 |, and multiplyingthe dual of Eq. (2.53), which since A = A† is given by a2 | A = a∗ a2 | , 2 (2.54)from the right by the ket-vector |a1 yield a2 | A |a1 = a1 a2 |a1 , (2.55) a2 | A |a1 = a∗ a2 |a1 . 2 (2.56)Thus, we have found that (a1 − a∗ ) a2 |a1 = 0 . 2 (2.57)The first part of the theorem is proven by employing the last result (2.57)for the case where |a1 = |a2 . Since |a1 is assumed to be a nonzero ket-vector one concludes that a1 = a∗ , namely a1 is real. Since this is true for 1any eigenvalue of A, one can rewrite Eq. (2.57) as (a1 − a2 ) a2 |a1 = 0 . (2.58)The second part of the theorem is proven by considering the case wherea1 = a2 , for which the above result (2.58) can hold only if a2 |a1 = 0.Namely eigenvectors corresponding to different eigenvalues are orthogonal. Consider a Hermitian operator A having a set of eigenvalues {an }n . Letgn be the degree of degeneracy of eigenvalue an , namely gn is the dimensionof the corresponding eigensubspace, which is denoted by Fn . For simplic-ity, assume that gn is finite for every n. Let {|an,1 , |an,2 , · · · , |an,gn } beEyal Buks Quantum Mechanics - Lecture Notes 24
  32. 32. 2.6. Observablesan orthonormal basis of the eigensubspace Fn , namely an,i′ |an,i = δ ii′ .Constructing such an orthonormal basis for Fn can be done by the so-calledGram-Schmidt process. Moreover, since eigenvectors of A corresponding todifferent eigenvalues are orthogonal, the following holds an′ ,i′ |an,i = δ nn′ δ ii′ , (2.59)In addition, all the ket-vectors |an,i are eigenvectors of A A |an,i = an |an,i . (2.60)Projectors. Projector operators are useful for expressing the properties ofan observable.Definition 2.6.4. An Hermitian operator P is called a projector if P 2 = P .Claim. The only possible eigenvalues of a projector are 0 and 1.Proof. Assume that |p is an eigenvector of P with an eigenvalue p, namelyP |p = p |p . Applying the operator P on both sides and using the fact thatP 2 = P yield P |p = p2 |p , thus p (1 − p) |p = 0, therefore since |p isassumed to be nonzero, either p = 0 or p = 1. A projector is said to project any given vector onto the eigensubspacecorresponding to the eigenvalue p = 1. Let {|an,1 , |an,2 , · · · , |an,gn } be an orthonormal basis of an eigensub-space Fn corresponding to an eigenvalue of an observable A. Such an ortho-normal basis can be used to construct a projection Pn onto Fn , which is givenby gn Pn = |an,i an,i | . (2.61) i=1 †Clearly, Pn is a projector since Pn = Pn and since gn gn 2 Pn = |an,i an,i |an,i′ an,i′ | = |an,i an,i | = Pn . (2.62) i,i′ =1 i=1 δii′Moreover, it is easy to show using the orthonormality relation (2.59) that thefollowing holds Pn Pm = Pm Pn = Pn δ nm . (2.63) For linear vector spaces of finite dimensionality, it can be shown that theset {|an,i }n,i forms a complete orthonormal basis of eigenvectors of a givenHermitian operator A. The generalization of this result for the case of ar-bitrary dimensionality is nontrivial, since generally such a set needs not beEyal Buks Quantum Mechanics - Lecture Notes 25
  33. 33. Chapter 2. State Vectors and Operatorscomplete. On the other hand, it can be shown that if a given Hermitian oper-ator A satisfies some conditions (e.g., A needs to be completely continuous)then completeness is guarantied. For all Hermitian operators of interest forthis course we will assume that all such conditions are satisfied. That is, forany such Hermitian operator A there exists a set of ket vectors {|an,i }, suchthat: 1. The set is orthonormal, namely an′ ,i′ |an,i = δ nn′ δ ii′ , (2.64) 2. The ket-vectors |an,i are eigenvectors, namely A |an,i = an |an,i , (2.65) where an ∈ R. 3. The set is complete, namely closure relation [see also Eq. (2.23)] is satis- fied gn 1= |an,i an,i | = Pn , (2.66) n i=1 n where gn Pn = |an,i an,i | (2.67) i=1 is the projector onto eigen subspace Fn with the corresponding eigenvalue an . The closure relation (2.66) can be used to express the operator A in termsof the projectors Pn gn gn A = A1 = A |an,i an,i | = an |an,i an,i | , (2.68) n i=1 n i=1that is A= an Pn . (2.69) n The last result is very useful when dealing with a function f (A) of theoperator A. The meaning of a function of an operator can be understood interms of the Taylor expansion of the function f (x) = fm xm , (2.70) mEyal Buks Quantum Mechanics - Lecture Notes 26
  34. 34. 2.6. Observableswhere 1 dm f fm = . (2.71) m! dxmWith the help of Eqs. (2.63) and (2.69) one finds that f (A) = fm Am m m = fm an Pn m n = fm am Pn n m n = fm am Pn , n n m f (an ) (2.72)thus f (A) = f (an ) Pn . (2.73) nExercise 2.6.1. Express the projector Pn in terms of the operator A andits set of eigenvalues.Solution 2.6.1. We seek a function f such that f (A) = Pn . Multiplyingfrom the right by a basis ket-vector |am,i yields f (A) |am,i = δ mn |am,i . (2.74)On the other hand f (A) |am,i = f (am ) |am,i . (2.75)Thus we seek a function that satisfy f (am ) = δ mn . (2.76)The polynomial function f (a) = K (a − am ) , (2.77) m=nwhere K is a constant, satisfies the requirement that f (am ) = 0 for everym = n. The constant K is chosen such that f (an ) = 1, that is a − am f (a) = , (2.78) an − am m=nEyal Buks Quantum Mechanics - Lecture Notes 27
  35. 35. Chapter 2. State Vectors and OperatorsThus, the desired expression is given by A − am Pn = . (2.79) m=n an − am2.7 Quantum MeasurementConsider a measurement of a physical variable denoted as A(c) performed ona quantum system. The standard textbook description of such a process isdescribed below. The physical variable A(c) is represented in quantum me-chanics by an observable, namely by a Hermitian operator, which is denotedas A. The correspondence between the variable A(c) and the operator A willbe discussed below in chapter 4. As we have seen above, it is possible to con-struct a complete orthonormal basis made of eigenvectors of the Hermitianoperator A having the properties given by Eqs. (2.64), (2.65) and (2.66). Inthat basis, the vector state |α of the system can be expressed as gn |α = 1 |α = an,i |α |an,i . (2.80) n i=1Even when the state vector |α is given, quantum mechanics does not gener-ally provide a deterministic answer to the question: what will be the outcomeof the measurement. Instead it predicts that: 1. The possible results of the measurement are the eigenvalues {an } of the operator A. 2. The probability pn to measure the eigen value an is given by gn pn = α| Pn |α = | an,i |α |2 . (2.81) i=1 Note that the state vector |α is assumed to be normalized. 3. After a measurement of A with an outcome an the state vector collapses onto the corresponding eigensubspace and becomes Pn |α |α → . (2.82) α| Pn |α It is easy to show that the probability to measure something is unityprovided that |α is normalized: pn = α| Pn |α = α| Pn |α = 1 . (2.83) n n nEyal Buks Quantum Mechanics - Lecture Notes 28
  36. 36. 2.8. Example - Spin 1/2We also note that a direct consequence of the collapse postulate is that twosubsequent measurements of the same observable performed one immediatelyafter the other will yield the same result. It is also important to note that theabove ’standard textbook description’ of the measurement process is highlycontroversial, especially, the collapse postulate. However, a thorough discus-sion of this issue is beyond the scope of this course. Quantum mechanics cannot generally predict the outcome of a specificmeasurement of an observable A, however it can predict the average, namelythe expectation value, which is denoted as A . The expectation value is easilycalculated with the help of Eq. (2.69) A = an pn = an α| Pn |α = α| A |α . (2.84) n n2.8 Example - Spin 1/2Spin is an internal degree of freedom of elementary particles. Electrons, forexample, have spin 1/2. This means, as we will see in chapter 6, that thestate of a spin 1/2 can be described by a state vector |α in a vector spaceof dimensionality 2. In other words, spin 1/2 is said to be a two-level system.The spin was first discovered in 1921 by Stern and Gerlach in an experimentin which the magnetic moment of neutral silver atoms was measured. Silveratoms have 47 electrons, 46 out of which fill closed shells. It can be shownthat only the electron in the outer shell contributes to the total magneticmoment of the atom. The force F acting on a magnetic moment µ moving ina magnetic field B is given by F = ∇ (µ · B). Thus by applying a nonuniformmagnetic field B and by monitoring the atoms’ trajectories one can measurethe magnetic moment. It is important to keep in mind that generally in addition to the spincontribution to the magnetic moment of an electron, also the orbital motionof the electron can contribute. For both cases, the magnetic moment is relatedto angular momentum by the gyromagnetic ratio. However this ratio takesdifferent values for these two cases. The orbital gyromagnetic ratio can beevaluated by considering a simple example of an electron of charge e movingin a circular orbit or radius r with velocity v. The magnetic moment is givenby AI µorbital = , (2.85) cwhere A = πr2 is the area enclosed by the circular orbit and I = ev/ (2πr)is the electrical current carried by the electron, thus erv µorbital = . (2.86) 2cEyal Buks Quantum Mechanics - Lecture Notes 29
  37. 37. Chapter 2. State Vectors and OperatorsThis result can be also written as µ µorbital = B L , (2.87)where L = me vr is the orbital angular momentum, and where e µB = (2.88) 2me cis the Bohr’s magneton constant. The proportionality factor µB / is theorbital gyromagnetic ratio. In vector form and for a more general case oforbital motion (not necessarily circular) the orbital gyromagnetic relation isgiven by µB µorbital = L. (2.89) On the other hand, as was first shown by Dirac, the gyromagnetic ratiofor the case of spin angular momentum takes twice this value 2µB µspin = S. (2.90)Note that we follow here the convention of using the letter L for orbitalangular momentum and the letter S for spin angular momentum. The Stern-Gerlach apparatus allows measuring any component of themagnetic moment vector. Alternatively, in view of relation (2.90), it can besaid that any component of the spin angular momentum S can be measured.The experiment shows that the only two possible results of such a measure-ment are + /2 and − /2. As we have seen above, one can construct a com-plete orthonormal basis to the vector space made of eigenvectors of any givenobservable. Choosing the observable Sz = S · ˆ for this purpose we construct za basis made of two vectors {|+; ˆ , |−; ˆ }. Both vectors are eigenvectors of z zSz Sz |+; ˆ = z |+; ˆ , z (2.91) 2 Sz |−; ˆ = − |−; ˆ . z z (2.92) 2In what follow we will use the more compact notation |+ = |+; ˆ , z (2.93) |− = |−; ˆ . z (2.94)The orthonormality property implied that + |+ = − |− = 1 , (2.95) − |+ = 0 . (2.96)The closure relation in the present case is expressed asEyal Buks Quantum Mechanics - Lecture Notes 30
  38. 38. 2.8. Example - Spin 1/2 |+ +| + |− −| = 1 . (2.97)In this basis any ket-vector |α can be written as |α = |+ + |α + |− − |α . (2.98)The closure relation (2.97) and Eqs. (2.91) and (2.92) yield Sz = (|+ +| − |− −|) (2.99) 2It is useful to define also the operators S+ and S− S+ = |+ −| , (2.100) S− = |− +| . (2.101)In chapter 6 we will see that the x and y components of S are given by Sx = (|+ −| + |− +|) , (2.102) 2 Sy = (−i |+ −| + i |− +|) . (2.103) 2All these ket-vectors and operators have matrix representation, which for thebasis {|+; ˆ , |−; ˆ } is given by z z 1 |+ = ˙ , (2.104) 0 0 |− = ˙ , (2.105) 1 01 Sx = ˙ , (2.106) 2 10 0 −i Sy = ˙ , (2.107) 2 i 0 1 0 Sz = ˙ , (2.108) 2 0 −1 01 S+ = ˙ , (2.109) 00 00 S− = ˙ . (2.110) 10Exercise 2.8.1. Given that the state vector of a spin 1/2 is |+; ˆ calculate z(a) the expectation values Sx and Sz (b) the probability to obtain a valueof + /2 in a measurement of Sx .Solution 2.8.1. (a) Using the matrix representation one hasEyal Buks Quantum Mechanics - Lecture Notes 31

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