Eyal Buks - Quantum Mechanics - Lecture Notes

Eyal Buks

Quantum Mechanics - LectureNotes

March 1, 2015

Technion

Preface

The dynamics of a quantum system is governed by the celebrated Schrdingerequation

id

dt| = H | , (0.1)

where i =1 and = 1.05457266 1034 J s is Plancks h-bar constant.

However, what is the meaning of the symbols | 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 | 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].

Contents

1. Hamiltons Formalism of Classical Physics . . . . . . . . . . . . . . . . 11.1 Action and Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Principle of Least Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Poissons Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2. State Vectors and Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1 Linear Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3 Diracs notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4 Dual Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5 Matrix Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.6 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.6.1 Hermitian Adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.6.2 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . 23

2.7 Quantum Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.8 Example - Spin 1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.9 Unitary Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.10 Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.11 Commutation Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.12 Simultaneous Diagonalization of Commuting Operators . . . . . 352.13 Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.14 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.15 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3. The Position and Momentum Observables . . . . . . . . . . . . . . . . 493.1 The One Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.1.1 Position Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.1.2 Momentum Representation . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2 Transformation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.3 Generalization for 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Contents

3.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4. Quantum Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.1 Time Evolution Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.2 Time Independent Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . 704.3 Example - Spin 1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.4 Connection to Classical Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 734.5 Symmetric Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5. The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.1 Eigenstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.2 Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.4 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6. Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1496.1 Angular Momentum and Rotation . . . . . . . . . . . . . . . . . . . . . . . . 1506.2 General Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1526.3 Simultaneous Diagonalization of J2 and Jz . . . . . . . . . . . . . . . . 1526.4 Example - Spin 1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1576.5 Orbital Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1576.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1636.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

7. Central Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2037.1 Simultaneous Diagonalization of the Operators H, L2 and Lz 2047.2 The Radial Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2067.3 Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2087.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2137.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

8. Density Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2238.1 Time Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2278.2 Quantum Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 2278.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2288.4 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

9. Time Independent Perturbation Theory . . . . . . . . . . . . . . . . . . 2939.1 The Level En . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

9.1.1 Nondegenerate Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2959.1.2 Degenerate Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

9.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2979.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

Eyal Buks Quantum Mechanics - Lecture Notes 6

Contents

9.4 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

10. Time-Dependent Perturbation Theory . . . . . . . . . . . . . . . . . . . . 33710.1 Time Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33710.2 Perturbation Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33810.3 Transition Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340

10.3.1 The Stationary Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34110.3.2 The Near-Resonance Case . . . . . . . . . . . . . . . . . . . . . . . . . 34310.3.3 H1 is Separable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

10.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34410.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

11. WKB Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35111.1 WKB Wavefunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35111.2 Turning Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35411.3 Bohr-Sommerfeld Quantization Rule . . . . . . . . . . . . . . . . . . . . . . 35811.4 Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36011.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36111.6 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

12. Path Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36912.1 Charged Particle in Electromagnetic Field . . . . . . . . . . . . . . . . . 36912.2 Classical Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37312.3 Aharonov-Bohm Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374

12.3.1 Two-slit Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37612.3.2 Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

12.4 One Dimensional Path Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 37912.4.1 One Dimensional Free Particle . . . . . . . . . . . . . . . . . . . . . 38012.4.2 Expansion Around the Classical Path . . . . . . . . . . . . . . . 38112.4.3 One Dimensional Harmonic Oscillator . . . . . . . . . . . . . . . 383

12.5 Semiclassical Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38712.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38812.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

13. Adiabatic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39713.1 Momentary Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39713.2 Gauge Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39913.3 Adiabatic Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39913.4 The Case of Two Dimensional Hilbert Space . . . . . . . . . . . . . . . 40013.5 Transition Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

13.5.1 The Case of Two Dimensional Hilbert Space . . . . . . . . . 40313.6 Slow and Fast Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40613.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40913.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410


Contents

14. The Quantized Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . 41514.1 Classical Electromagnetic Cavity . . . . . . . . . . . . . . . . . . . . . . . . . 41514.2 Quantum Electromagnetic Cavity . . . . . . . . . . . . . . . . . . . . . . . . 42014.3 Periodic Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 42214.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42314.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424

15. Light Matter Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43115.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43115.2 Transition Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432

15.2.1 Spontaneous Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43215.2.2 Stimulated Emission and Absorption . . . . . . . . . . . . . . . . 43315.2.3 Selection Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434

15.3 Semiclassical Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43615.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43815.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438

16. Identical Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44516.1 Basis for the Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44516.2 Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44816.3 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44916.4 Changing the Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45116.5 Many Particle Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453

16.5.1 One-Particle Observables . . . . . . . . . . . . . . . . . . . . . . . . . . 45316.5.2 Two-Particle Observables . . . . . . . . . . . . . . . . . . . . . . . . . . 454

16.6 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45516.7 Momentum Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45816.8 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46016.9 The Electron Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46116.10Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46316.11Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

17. Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48317.1 Macroscopic Wavefunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483

17.1.1 Single Particle in Electromagnetic Field . . . . . . . . . . . . . 48317.1.2 Drude Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48517.1.3 The Macroscopic Quantum Model . . . . . . . . . . . . . . . . . . 48917.1.4 London Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

17.2 The Josephson Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49317.2.1 The First Josephson Relation . . . . . . . . . . . . . . . . . . . . . . 49317.2.2 The Second Josephson Relation . . . . . . . . . . . . . . . . . . . . 49417.2.3 The Energy of a Josephson Junction . . . . . . . . . . . . . . . . 495

17.3 RF SQUID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49617.3.1 Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49717.3.2 Flux Quantum Bit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499


Contents

17.3.3 Qubit Readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50017.4 BCS Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

17.4.1 Phonon Mediated Electron-Electron Interaction . . . . . . 50917.4.2 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51217.4.3 Bogoliubov Transformation . . . . . . . . . . . . . . . . . . . . . . . . 51317.4.4 The Energy Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51617.4.5 The Ground State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51817.4.6 Pairing Wavefunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520

17.5 The Josephson Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52117.5.1 The Second Josephson Relation . . . . . . . . . . . . . . . . . . . . 52217.5.2 The Energy of a Josephson Junction . . . . . . . . . . . . . . . . 52417.5.3 The First Josephson Relation . . . . . . . . . . . . . . . . . . . . . . 527


18. Open Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54118.1 Classical Resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54118.2 Quantum Resonator Coupled to Thermal Bath . . . . . . . . . . . . . 542

18.2.1 The closed System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54218.2.2 Coupling to Thermal Bath . . . . . . . . . . . . . . . . . . . . . . . . 54318.2.3 Thermal Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546

18.3 Two Level System Coupled to Thermal Bath . . . . . . . . . . . . . . . 54918.3.1 The Closed System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54918.3.2 Coupling to Thermal Baths . . . . . . . . . . . . . . . . . . . . . . . . 55018.3.3 Thermal Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55418.3.4 Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55518.3.5 The Bloch Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557


References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567


1. Hamiltons Formalism of Classical Physics

In this chapter the Hamiltons formalism of classical physics is introduced,with a special emphasis on the concepts that are needed for quantum me-chanics.

1.1 Action and Lagrangian

Consider 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

S =

t2t1

dt L , (1.4)

where L is the Lagrangian function of the system. In general, the Lagrangianis 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 on the trajectory taken by the system from point Q1 at time t1

Chapter 1. Hamiltons Formalism of Classical Physics

t

Q

t1 t2

Q2

Q1

t

Q

t1 t2

Q2

Q1

Fig. 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 the timedependency is denoted as

Q (t) = Q (t) . (1.7)

1.2 Principle of Least Action

For 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 Q (t). The trajectory Q (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 Q (t).

In a weaker version of this principle, the action obtains a local minimumfor the trajectory Q (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 Q (t), for which the action obtainsits minimum value, obeys the Euler-Lagrange equations of motion, which aregiven by


1.2. Principle of Least Action

d

dt

Lqn

=Lqn

, (1.8)

where n = 1, 2, , N.Proof. Consider another trajectory Q (t) from point Q1 at time t1 to pointQ2 at time t2 (see Fig. 1.2). The difference

Q = Q (t)Q (t) = (q1, q2, , qN) (1.9)

is assumed to be infinitesimally small. To lowest order in Q the change inthe action S is given by

S =

t2t1

dt L

=

t2t1

dt

[Nn=1

Lqn

qn +Nn=1

Lqn

qn

]

=

t2t1

dt

[Nn=1

Lqn

qn +Nn=1

Lqn

d

dtqn

].

(1.10)

Integrating the second term by parts leads to

S =

t2t1

dtNn=1

(Lqn

ddt

Lqn

)qn

+Nn=1

[Lqn

qn

t2t1

.

(1.11)

The last term vanishes since

Q (t1) = Q (t2) = 0 . (1.12)

The principle of least action implies that

S = 0 . (1.13)

This has to be satisfied for any Q, therefore the following must hold

d

dt

Lqn

=Lqn

. (1.14)



t

Q

t1 t2

Q2

Q1

t

Q

t1 t2

Q2

Q1

Fig. 1.2. The classical trajectory Q (t) and the trajectory Q (t).

In what follows we will assume for simplicity that the kinetic energy T ofthe 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

Fn = Uqn

+d

dt

U

qn. (1.15)

When the potential energy can be expressed as a function of the coordinatesQ 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

L = T U = mx2

2 U (x) . (1.17)

From the Euler-Lagrange equation

d

dt

Lx

=Lx

, (1.18)

one finds that

mx = Ux

. (1.19)


1.3. Hamiltonian

1.3 Hamiltonian

The 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

pn =Lqn

. (1.20)

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

H =Nn=1

pnqn L , (1.22)

where L is the Lagrangian.Theorem 1.3.1. The classical trajectory satisfies the Hamilton-Jacobi equa-tions of motion, which are given by

qn =Hpn

, (1.23)

pn = Hqn

, (1.24)

where n = 1, 2, , N.Proof. The differential of H is given by

dH = dNn=1

pnqn dL

=Nn=1

qndpn + pndqn Lqnddt

Lqn

dqn Lqnpn

dqn

Lt dt

=Nn=1

(qndpn pndqn) Ltdt .

(1.25)



Thus the following holds

qn =Hpn

, (1.26)

pn = Hqn

, (1.27)

Lt=

Ht

. (1.28)

Corollary 1.3.1. The following holds

dHdt

=Ht

. (1.29)

Proof. Using Eqs. (1.23) and (1.24) one finds that

dHdt

=Nn=1

(Hqn

qn +Hpn

pn

)

=0

+Ht

=Ht

. (1.30)

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 =n,m

nmqnqm , (1.31)

where nm 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

pl =Lql

=T

ql, (1.33)

where L = T U is the Lagrangian. The Hamiltonian is thus given by


1.4. Poissons Brackets

H =Nl=1

plql L

=l

T

qlql (T U)

=l,n,m

nm

qmqnqlnl

+ qnqmqlml

ql T + U= 2

n,m

nmqnqm T

T + U

= T + U .

(1.34)

1.4 Poissons Brackets

Consider 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 Poissons brackets are defined by

{F,G} =Nn=1

(F

qn

G

pn Fpn

G

qn

), (1.37)

The Poissons 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

dt= {F,H}+ F

t. (1.38)

Proof. Using Eqs. (1.23) and (1.24) one finds that the time derivative of Fis given by



dF

dt=

Nn=1

(F

qnqn +

F

pnpn

)+F

t

=Nn=1

(F

qn

Hpn

Fpn

Hqn

)+F

t

= {F,H}+ Ft

.

(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

dt= 0 . (1.40)

1.5 Problems

1. Consider a particle having charge q and mass m in electromagnetic fieldcharacterized by the scalar potential and the vector potential A. Theelectric field E and the magnetic field B are given by

E = 1c

A

t, (1.41)

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 givenby

L = 12mr2 q+ q

cA r , (1.43)

by showing that the corresponding Euler-Lagrange equations areequivalent to Newtons 2nd law (i.e., F = mr).

b) Show that the Hamilton-Jacobi equations are equivalent to Newtons2nd law.

c) Gauge transformation The electromagnetic field is invariant un-der the gauge transformation of the scalar and vector potentials

A A+ , (1.44) 1

c

t(1.45)

where = (r, t) is an arbitrary smooth and continuous functionof r and t. What effect does this gauge transformation have on theLagrangian and Hamiltonian? Is the motion affected?


1.6. Solutions

L CL C

Fig. 1.3. LC resonator.

2. Consider an LC resonator made of a capacitor having capacitance C inparallel with an inductor having inductance L (see Fig. 1.3). The stateof the system is characterized by the coordinate q , which is the chargestored 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} = 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,FG} = {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 + y2 + z2 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 givenby

L = 12mr2 q+ q

cA r . (1.54)



a) The Euler-Lagrange equation for the coordinate x is given by

d

dt

Lx

=Lx

, (1.55)

where

d

dt

Lx

= mx+q

c

(Axt

+ xAxx

+ yAxy

+ zAxz

), (1.56)

and

Lx

= q x+q

c

(xAxx

+ yAyx

+ zAzx

), (1.57)

thus

mx = q x

qc

Axt

qEx

+q

c

y

(Ayx

Axy

)

(A)z

z

(Axz

Azx

)

(A)y (r(A))x

,

(1.58)or

mx = qEx +q

c(rB)x . (1.59)

Similar equations are obtained for y and z in the same way. These 3equations can be written in a vector form as

mr = q

(E+

1

crB

). (1.60)

b) The variable vector canonically conjugate to the coordinates vectorr is given by

p =Lr

=mr+q

cA . (1.61)

The Hamiltonian is thus given by


1.6. Solutions

H = p rL= r

(p 1

2mr q

cA

)+ q

=1

2mr2 + q

=

(p qcA

)22m

+ q .

(1.62)The Hamilton-Jacobi equation for the coordinate x is given by

x =Hpx

, (1.63)

thus

x =pxqcAx

m, (1.64)

or

px =mx+q

cAx . (1.65)

The Hamilton-Jacobi equation for the canonically conjugate variablepx is given by

px = Hx

, (1.66)

where

px =mx+q

c

(xAxx

+ yAxy

+ zAxz

)+q

c

Axt

, (1.67)

and

Hx

=q

c

(pxqcAx

m

Axx

+pyqcAy

m

Ayx

+pzqcAz

m

Azx

) q

x

=q

c

(xAxx

+ yAyx

+ zAzx

) q

x,

(1.68)thus

mx = q x

qc

Axt

+q

c

[y

(Ayx

Axy

) z

(Axz

Azx

)].

(1.69)

The last result is identical to Eq. (1.59).



c) Clearly, the fields E and B, which are given by Eqs. (1.41) and (1.42)respectively, are unchanged since

(

t

) ()

t= 0 , (1.70)

and

() = 0 . (1.71)Thus, even though both L and H are modified, the motion, whichdepends on E and B only, is unaffected.

2. The kinetic energy in this case T = Lq2/2 is the energy stored in theinductor, and the potential energy U = q2/2C is the energy stored in thecapacitor.

a) The Lagrangian is given by

L = T U = Lq2

2 q

2

2C. (1.72)

The Euler-Lagrange equation for the coordinate q is given by

d

dt

Lq

=Lq

, (1.73)

thus

Lq +q

C= 0 . (1.74)

This equation expresses the requirement that the voltage across thecapacitor is the same as the one across the inductor.

b) The canonical conjugate momentum is given by

p =Lq

= Lq , (1.75)

and the Hamiltonian is given by

H = pq L = p2

2L+

q2

2C. (1.76)

Hamilton-Jacobi equations read

q =p

L(1.77)

p = qC, (1.78)

thus

Lq +q

C= 0 . (1.79)


1.6. Solutions

c) Using the definition (1.37) one has

{q, p} = qq

p

p qp

p

q= 1 . (1.80)

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 coordinatestransformation

xa = xa (q1, q2, ..., t) , (1.81)

where a = 1, 2, . The following holds

xa =xaqb

qb +xat

, (1.82)

where the summation convention is being used, namely, repeated indicesare summed over. Moreover

Lqa

=Lxb

xbqa

+Lxb

xbqa

, (1.83)

and

d

dt

(Lqa

)=d

dt

(Lxb

xbqa

). (1.84)

As can be seen from Eq. (1.82), one has

xbqa

=xbqa

. (1.85)

Thus, using Eqs. (1.83) and (1.84) one finds

d

dt

(Lqa

) Lqa

=d

dt

(Lxb

xbqa

) Lxb

xbqa

Lxb

xbqa

=

[d

dt

(Lxb

) Lxb

]xbqa

+

[d

dt

(xbqa

) xbqa

]Lxb

.

(1.86)

As can be seen from Eq. (1.82), the second term vanishes since



xbqa

=2xbqaqc

qc +2xbtqa

=d

dt

(xbqa

),

thus

d

dt

(Lqa

) Lqa

=

[d

dt

(Lxb

) Lxb

]xbqa

. (1.87)

The last result shows that if the coordinate transformation is reversible,namely if det (xb/qa) = 0 then Lagrange equations are coordinateinvariant.

5. The angular momentum L is given by

L = r p = det x y zx y zpx py pz

, (1.88)where r = (x, y, z) is the position vector and where p = (px, py, pz) is themomentum vector. The Hamiltonian is given by

H = p2

2m+ V (r) . (1.89)

Using

{xi, pj} = ij , (1.90)Lz = xpy ypx , (1.91)

one finds that{p2, Lz

}={p2x, Lz

}+{p2y, Lz

}+{p2z, Lz

}={p2x, xpy

} {p2y, ypx}= 2pxpy + 2pypx= 0 ,

(1.92)

and{r2, Lz

}={x2, Lz

}+{y2, Lz

}+{z2, Lz

}= y {x2, px}+ {y2, 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.


2. State Vectors and Operators

In quantum mechanics the state of a physical system is described by a statevector |, which is a vector in a vector space F , namely

| F . (2.1)Here, we have employed the Diracs ket-vector notation | 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 Space

A linear vector space F is a set {|} 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. |+ | = |+ | F for every | F and | F2. c | = | c F for every | 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 | , | F is denoted as |. Theinner product is a function F2 C that satisfies the following properties:

Chapter 2. State Vectors and Operators

| C , (2.2)

| = | , (2.3) (c1 |1+ c2 |2) = c1 |1+ c2 |2 , where c1, c2 C , (2.4)

| R and | 0. Equality holds iff | = 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 | is called the norm of the vector | F . A normalized vector has a unity norm, namely | = 1. Every nonzero vector 0 = | F can be normalized using the transfor-mation

| | | . (2.6) The vectors | F and | F are said to be orthogonal if | = 0. A set of vectors {|n}n, where |n F is called a complete orthonormalbasis if

The vectors are all normalized and orthogonal to each other, namely

m |n = nm . (2.7)

Every | F can be written as a superposition of the basis vectors,namely

| =n

cn |n , (2.8)

where cn C. By evaluating the inner product m |, where | is given by Eq. (2.8)one finds with the help of Eq. (2.7) and property (2.4) of inner productsthat

m | = m(

n

cn |n)=n

cnm |n =nm

= cm . (2.9)

The last result allows rewriting Eq. (2.8) as

| =n

cn |n =n

|n cn =n

|n n | . (2.10)


2.3. Diracs notation

2.2 Operators

Operators, 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 | F for every | F .

Some important definitions and comments are listed below:

The operators X : F F and Y : F F are said to be equal, namelyX = Y , if for every | F the following holds

X | = Y | . (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 ifA (c1 |1+ c2 |2) = c1A |1+ c2A |2 (2.14)

for every |1 , |2 F and c1, c2 C. The operators X : F F and Y : F F can be multiplied, where

XY | = X (Y |) (2.15)for any | F .

Operator multiplication is associativeX (Y Z) = (XY )Z = XYZ . (2.16)

However, in general operator multiplication needs not be commutativeXY = YX . (2.17)

2.3 Diracs notation

In Diracs notation the inner product is considered as a multiplication of twomathematical objects called bra and ket

| = |bra

|ket

. (2.18)

While the ket-vector | is a vector in F , the bra-vector | represents afunctional that maps any ket-vector | F to the complex number |.



While 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

A = | | . (2.19)

The outer product A is clearly an operator since for any | F the objectA | is a ket-vector

A | = (| |) | = | | C

F . (2.20)

Moreover, according to property (2.4),A is linear since for every |1 , |2 F and c1, c2 C the following holds

A (c1 |1+ c2 |2) = | | (c1 |1+ c2 |2)= | (c1 |1+ c2 |2)= c1A |1+ c2A |2 .

(2.21)

With Diracs notation Eq. (2.10) can be rewritten as

| =(

n

|n n|)| . (2.22)

Since the above identity holds for any | F one concludes that the quantityin brackets is the identity operator, which is denoted as 1, namely

1 =n

|n n| . (2.23)

This result, which is called the closure relation, implies that any completeorthonormal basis can be used to express the identity operator.

2.4 Dual Correspondence

As we have mentioned above, the bra-vector | represents a functional map-ping any ket-vector | F to the complex number |. Moreover, sincethe inner product is linear [see property (2.4) above], such a mapping is linear,namely for every |1 , |2 F and c1, c2 C the following holds

| (c1 |1+ c2 |2) = c1 |1+ c2 |2 . (2.24)

The set of linear functionals from F to C, namely, the set of functionals F : F C that satisfy


2.4. Dual Correspondence

F (c1 |1+ c2 |2) = c1F (|1) + c2F (|2) (2.25)

for every |1 , |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

| | , (2.26)

where | F and | F. What is the dual of the ket-vector | =c1 |1+ c2 |2, where |1 , |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 | for an arbitrary ket-vector | F

| = |= (c1 |1+ c2 |2)= c1 1 |+ c2 2 |= (c1 1|+ c2 2|) | .

(2.27)

From this result we conclude that the duality relation takes the form

c1 1|+ c2 2| c1 |1+ c2 |2 . (2.28)

The last relation describes how to map any given ket-vector | Fto its dual F = | : F C, where F F is a linear functional thatmaps any ket-vector | F to the complex number |. 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 =n

(F (|n)) |n . (2.29)

The duality is demonstrated by proving the two claims below:

Claim. |DD = | for any | F , where |DD is the dual of the dual of|.Proof. The dual of | is the bra-vector |, whereas the dual of | is foundusing Eqs. (2.29) and (2.23), thus



|DD =n

|n =n |

|n

=n

|n n |

=n

|n n| =1

|

= | .(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 Fof |FD is given by

FDD =n

F (|n) n| . (2.31)

Consider an arbitrary ket-vector | F that is written as a superpositionof the complete orthonormal basis vectors, namely

| =m

cm |m . (2.32)

Using the above expression for FDD and the linearity property one finds that

FDD | =n,m

cmF (|n) n |m mn

=n

cnF (|n)

= F

(n

cn |n)

= F (|) ,(2.33)

therefore, FDD = F .

2.5 Matrix Representation

Given 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).


2.5. Matrix Representation

The inner product between the bra-vector | and the ket-vector | canbe written as

| = | 1 |=n

|n n|

=( |1 |2 )

1 |2 |...

.(2.34)

Thus, the inner product can be viewed as a product between the row vector

| = ( |1 |2 ) , (2.35)which is the matrix representation of the bra-vector |, and the columnvector

| =

1 |2 |...

, (2.36)which is the matrix representation of the ket-vector |. Obviously, bothrepresentations are basis dependent.

Multiplying the relation | = X | from the right by the basis bra-vectorm| and employing again the closure relation (2.23) yields

m | = m|X | = m|X1 | =n

m|X |n n | , (2.37)

or in matrix form1 |2 |...

=1|X |1 1|X |2 2|X |1 2|X |2

......

1 |2 |

...

. (2.38)In view of this expression, the matrix representation of the linear operatorX is given by

X=

1|X |1 1|X |2 2|X |1 2|X |2 ...

...

. (2.39)Alternatively, the last result can be written as

Xnm = n|X |m , (2.40)where Xnm is the element in row n and column m of the matrix represen-tation of the operator X.



Such matrix representation of linear operators can be useful also for mul-tiplying linear operators. The matrix elements of the product Z = XY aregiven by

m|Z |n = m|XY |n = m|X1Y |n =l

m|X |l l|Y |n .

(2.41)

Similarly, the matrix representation of the outer product | | is givenby

| | =

1 |2 |...

( |1 |2 )

=

1 | |1 1 | |2 2 | |1 2 | |2 ...

...

.(2.42)

2.6 Observables

Measurable physical variables are represented in quantum mechanics by Her-mitian operators.

2.6.1 Hermitian Adjoint

Definition 2.6.1. The Hermitian adjoint of an operator X is denoted as X

and is defined by the following duality relation

|X X | . (2.43)Namely, for any ket-vector | F , the dual to the ket-vector X | is thebra-vector |X.Definition 2.6.2. An operator is said to be Hermitian if X = X.

Below we prove some simple relations:

Claim. |X | = |X |

Proof. Using the general property (2.3) of inner products one has

|X | = | (X |) = ((|X) |) = |X | . (2.44)Note that this result implies that if X = X then |X | = |X |.


2.6. Observables

Claim.(X)= X

Proof. For any | , | F the following holds

|X | = (|X |) = |X | = | (X) | , (2.45)thus

(X)= X.

Claim. (XY ) = Y X

Proof. Applying XY on an arbitrary ket-vector | F and employing theduality correspondence yield

XY | = X (Y |) (|Y )X = |Y X , (2.46)thus

(XY ) = Y X . (2.47)

Claim. If X = | | then X = | |Proof. By applying X on an arbitrary ket-vector | F and employing theduality correspondence one finds that

X | = (| |) | = | ( |) ( |) | = | | = |X ,(2.48)

where X = | |.

2.6.2 Eigenvalues and Eigenvectors

Each 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)



Clearly, Fn is closed under vector addition and scalar multiplication, namelyc1 |1+c2 |2 Fn for every |1, |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 = dimFn . (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 = a2 a2| , (2.54)from the right by the ket-vector |a1 yield

a2|A |a1 = a1 a2 |a1 , (2.55)a2|A |a1 = a2 a2 |a1 . (2.56)

Thus, we have found that

(a1 a2) a2 |a1 = 0 . (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 = a1, namely a1 is real. Since this is true forany 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} be


2.6. Observables

an orthonormal basis of the eigensubspace Fn, namely an,i |an,i = 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 = nnii , (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

Pn =

gni=1

|an,i an,i| . (2.61)

Clearly, Pn is a projector since P n = Pn and since

P 2n =

gni,i=1

|an,i an,i |an,i ii

an,i | =gni=1

|an,i an,i| = Pn . (2.62)

Moreover, it is easy to show using the orthonormality relation (2.59) that thefollowing holds

PnPm = PmPn = Pnnm . (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 be



complete. 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 = nnii , (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

1 =n

gni=1

|an,i an,i| =n

Pn , (2.66)

where

Pn =

gni=1

|an,i an,i| (2.67)

is the projector onto eigen subspace Fn with the corresponding eigenvaluean.

The closure relation (2.66) can be used to express the operator A in termsof the projectors Pn

A = A1 =n

gni=1

A |an,i an,i| =n

an

gni=1

|an,i an,i| , (2.68)

that is

A =n

anPn . (2.69)

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) =m

fmxm , (2.70)


2.6. Observables

where

fm =1

m!

dmf

dxm. (2.71)

With the help of Eqs. (2.63) and (2.69) one finds that

f (A) =m

fmAm

=m

fm

(n

anPn

)m=m

fmn

amn Pn

=n

m

fmamn

f(an)

Pn ,

(2.72)

thus

f (A) =n

f (an)Pn . (2.73)

Exercise 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 = 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) = mn . (2.76)

The polynomial function

f (a) = Km=n

(a am) , (2.77)

where 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

f (a) =m=n

a aman am , (2.78)



Thus, the desired expression is given by

Pn =m=n

A aman am . (2.79)

2.7 Quantum Measurement

Consider 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 | of the system can be expressed as

| = 1 | =n

gni=1

an,i | |an,i . (2.80)

Even when the state vector | 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 theoperator A.

2. The probability pn to measure the eigen value an is given by

pn = |Pn | =gni=1

|an,i ||2 . (2.81)

Note that the state vector | 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 ||Pn | . (2.82)It is easy to show that the probability to measure something is unity

provided that | is normalized:n

pn =n

|Pn | = |(

n

Pn

)| = 1 . (2.83)


2.8. Example - Spin 1/2

We 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 =n

anpn =n

an |Pn | = |A | . (2.84)

2.8 Example - Spin 1/2

Spin 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 | 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 moving ina magnetic field B is given by F = ( 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

orbital =AI

c, (2.85)

where A = r2 is the area enclosed by the circular orbit and I = ev/ (2r)is the electrical current carried by the electron, thus

orbital =erv

2c. (2.86)



This result can be also written as

orbital =BL , (2.87)

where L = mevr is the orbital angular momentum, and where

B =e

2mec(2.88)

is the Bohrs magneton constant. The proportionality factor B/ is theorbital gyromagnetic ratio. In vector form and for a more general case oforbital motion (not necessarily circular) the orbital gyromagnetic relation isgiven by

orbital =BL . (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

spin =2BS . (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 z for this purpose we constructa basis made of two vectors {|+; z , |; z}. Both vectors are eigenvectors ofSz

Sz |+; z = 2|+; z , (2.91)

Sz |; z = 2|; z . (2.92)

In what follow we will use the more compact notation

|+ = |+; z , (2.93)| = |; z . (2.94)

The orthonormality property implied that

+ |+ = | = 1 , (2.95) |+ = 0 . (2.96)

The closure relation in the present case is expressed as


2.8. Example - Spin 1/2

|+ +|+ | | = 1 . (2.97)

In this basis any ket-vector | can be written as

| = |+ + |+ | | . (2.98)

The closure relation (2.97) and Eqs. (2.91) and (2.92) yield

Sz =

2(|+ +| | |) (2.99)

It 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(|+ |+ | +|) , (2.102)

Sy =

2(i |+ |+ i | +|) . (2.103)

All these ket-vectors and operators have matrix representation, which for thebasis {|+; z , |; z} is given by

|+ =(10

), (2.104)

| =(01

), (2.105)

Sx =

2

(0 11 0

), (2.106)

Sy =

2

(0 ii 0

), (2.107)

Sz =

2

(1 00 1

), (2.108)

S+ =

(0 10 0

), (2.109)

S = (0 01 0

). (2.110)

Exercise 2.8.1. Given that the state vector of a spin 1/2 is |+; z calculate(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 has



Sx = +|Sx |+ = 2

(1 0)(0 11 0

)(10

)= 0 , (2.111)

Sz = +|Sz |+ = 2

(1 0)(1 00 1

)(10

)=

2. (2.112)

(b) First, the eigenvectors of the operator Sx are found by solving the equa-tion Sx | = |, which is done by diagonalization of the matrix represen-tation of Sx. The relation Sx | = | for the two eigenvectors is writtenin a matrix form as

2

(0 11 0

)( 1212

)=

2

(1212

), (2.113)

2

(0 11 0

)( 12

12

)=

2

(12

12

). (2.114)

That is, in ket notation

Sx |; x = 2|; x , (2.115)

where the eigenvectors of Sx are given by

|; x = 12(|+ |) . (2.116)

Using this result the probability p+ is easily calculated

p+ = |+ |+; x|2 =+| 12 (|++ |)

2 = 12 . (2.117)2.9 Unitary Operators

Unitary operators are useful for transforming from one orthonormal basis toanother.

Definition 2.9.1. An operator U is said to be unitary if U = U1, namelyif UU = UU = 1.

Consider two observables A and B, and two corresponding complete andorthonormal bases of eigenvectors

A |an = an |an , am |an = nm,n

|an an| = 1 , (2.118)

B |bn = bn |bn , bm |bn = nm,n

|bn bn| = 1 . (2.119)

The operator U , which is given by


2.10. Trace

U =n

|bn an| , (2.120)

transforms each of the basis vector |an to the corresponding basis vector |bnU |an = |bn . (2.121)

It is easy to show that the operator U is unitary

UU =n,m

|an bn |bm nm

am| =n

|an an| = 1 . (2.122)

The matrix elements of U in the basis {|an} are given byan|U |am = an |bm , (2.123)

and those of U by

an|U |am = bn |am .Consider a ket vector

| =n

|an an | , (2.124)

which can be represented as a column vector in the basis {|an}. The nthelement of such a column vector is an |. The operator U can be employedfor finding the corresponding column vector representation of the same ket-vector | in the other basis {|bn}

bn | =m

bn |am am | =m

an|U |am am | . (2.125)

Similarly, Given an operator X the relation between the matrix elementsan|X |am in the basis {|an} to the matrix elements bn|X |bm in thebasis {|bn} is given by

bn|X |bm =k,l

bn |ak ak|X |al al |bm

=k,l

an|U |ak ak|X |al al|U |am .

(2.126)

2.10 Trace

Given an operator X and an orthonormal and complete basis {|an}, thetrace of X is given by



Tr (X) =n

an|X |an . (2.127)

It is easy to show that Tr (X) is independent on basis, as is shown below:

Tr (X) =n

an|X |an

=n,k,l

an |bk bk|X |bl bl |an

=n,k,l

bl |an an |bk bk|X |bl

=k,l

bl |bk kl

bk|X |bl

=k

bk|X |bk .

(2.128)

The proof of the following relation

Tr (XY ) = Tr (Y X) , (2.129)

is left as an exercise.

2.11 Commutation Relation

The commutation relation of the operators A and B is defined as

[A,B] = AB BA . (2.130)As an example, the components Sx, Sy and Sz of the spin angular momentumoperator, satisfy the following commutation relations

[Si, Sj ] = iijkSk , (2.131)

where

ijk =

0 i, j, k are not all different1 i, j, k is an even permutation of x, y, z1 i, j, k is an odd permutation of x, y, z

(2.132)

is the Levi-Civita symbol. Equation (2.131) employs the Einsteins conven-tion, according to which if an index symbol appears twice in an expression,it is to be summed over all its allowed values. Namely, the repeated index kshould be summed over the values x, y and z:

ijkSk = ijxSx + ijySy + ijzSz . (2.133)


2.12. Simultaneous Diagonalization of Commuting Operators

Moreover, the following relations hold

S2x = S2y = S

2z =

1

42 , (2.134)

S2 = S2x + S2y + S

2z =

3

42 . (2.135)

The relations below, which are easy to prove using the above definition,are very useful for evaluating commutation relations

[F,G] = [G,F ] , (2.136)[F,F ] = 0 , (2.137)

[E + F,G] = [E,G] + [F,G] , (2.138)

[E,FG] = [E,F ]G+ F [E,G] . (2.139)

2.12 Simultaneous Diagonalization of CommutingOperators

Consider an observable A having a set of eigenvalues {an}. Let gn be thedegree of degeneracy of eigenvalue an, namely gn is the dimension of thecorresponding eigensubspace, which is denoted by Fn. Thus the followingholds

A |an,i = an |an,i , (2.140)

where i = 1, 2, , gn, and

an,i |an,i = nnii . (2.141)

The set of vectors {|an,1 , |an,2 , , |an,gn} forms an orthonormal basis forthe eigensubspace Fn. The closure relation can be written as

1 =n

gni=1

|an,i an,i| =n

Pn , (2.142)

where

Pn =

gni=1

|an,i an,i| . (2.143)

Now consider another observable B, which is assumed to commute withA, namely [A,B] = 0.

Claim. The operator B has a block diagonal matrix in the basis {|an,i},namely am,j |B |an,i = 0 for n =m.



Proof. Multiplying Eq. (2.140) from the left by am,j |B yields

am,j |BA |an,i = an am,j |B |an,i . (2.144)

On the other hand, since [A,B] = 0 one has

am,j |BA |an,i = am,j |AB |an,i = am am,j |B |an,i , (2.145)

thus

(an am) am,j |B |an,i = 0 . (2.146)

For a given n, the gn gn matrix an,i |B |an,i is Hermitian, namelyan,i |B |an,i = an,i|B |an,i. Thus, there exists a unitary transformationUn, which maps Fn onto Fn, and which diagonalizes the block of B in thesubspace Fn. Since Fn is an eigensubspace of A, the block matrix of A in thenew basis remains diagonal (with the eigenvalue an). Thus, we conclude thata complete and orthonormal basis of common eigenvectors of both operatorsA and B exists. For such a basis, which is denoted as {|n,m}, the followingholds

A |n,m = an |n,m , (2.147)B |n,m = bm |n,m . (2.148)

2.13 Uncertainty Principle

Consider a quantum system in a state |n,m, which is a common eigenvectorof the commuting observables A and B. The outcome of a measurement ofthe observable A is expected to be an with unity probability, and similarly,the outcome of a measurement of the observable B is expected to be bmwith unity probability. In this case it is said that there is no uncertaintycorresponding to both of these measurements.

Definition 2.13.1. The variance in a measurement of a given observable A

of a quantum system in a state | is given by(A)

2, where A = AA,

namely(A)2

=A2 2A A+ A2

=A2 A2 , (2.149)

where

A = |A | , (2.150)A2= |A2 | . (2.151)


2.13. Uncertainty Principle

Example 2.13.1. Consider a spin 1/2 system in a state | = |+; z. UsingEqs. (2.99), (2.102) and (2.134) one finds that

(Sz)2=S2z Sz2 = 0 , (2.152)

(Sx)2=S2x Sx2 = 1

42 . (2.153)

The last example raises the question: can one find a state | for whichthe variance in the measurements of both Sz and Sx vanishes? According tothe uncertainty principle the answer is no.

Theorem 2.13.1. The uncertainty principle - Let A and B be two observ-ables. For any ket-vector | the following holds

(A)2(B)2

14|[A,B]|2 . (2.154)

Proof. Applying the Schwartz inequality [see Eq. (2.166)], which is given by

u |u v |v |u |v|2 , (2.155)for the ket-vectors

|u = A | , (2.156)|v = B | , (2.157)

and exploiting the fact that (A) = A and (B) = B yield(A)2

(B)2

|AB|2 . (2.158)

The term AB can be written as

AB =1

2[A,B] +

1

2[A,B]+ , (2.159)

where

[A,B] = AB BA , (2.160)[A,B]+ = AB +BA . (2.161)

While the term [A,B] is anti-Hermitian, whereas the term [A,B]+ isHermitian, namely

([A,B]) = (AB BA) = BAAB = [A,B] ,([A,B]+

)= (AB +BA)

= BA+AB = [A,B]+ .

In general, the following holds

|X | = |X | ={ |X | if X is Hermitian|X | if X is anti-Hermitian , (2.162)



thus

AB = 12[A,B]

I

+1

2

[A,B]+

R

, (2.163)

and consequently

|AB|2 = 14|[A,B]|2 + 1

4

[A,B]+2 . (2.164)Finally, with the help of the identity [A,B] = [A,B] one finds that

(A)2(B)2

14|[A,B]|2 . (2.165)

2.14 Problems

1. Derive the Schwartz inequality

|u |v| u |u

v |v , (2.166)

where |u and |v are any two vectors of a vector space F .2. Derive the triangle inequality:

(u|+ v|) (|u+ |v) u |u+

v |v . (2.167)

3. Show that if a unitary operator U can be written in the form U = 1+iF ,where is a real infinitesimally small number, then the operator F isHermitian.

4. A Hermitian operator A is said to be positive-definite if, for any vector|u, u|A |u 0. Show that the operator A = |a a| is Hermitian andpositive-definite.

5. Show that if A is a Hermitian positive-definite operator then the followinghold

|u|A |v| u|A |u

v|A |v . (2.168)

6. Find the expansion of the operator (A B)1 in a power series in ,assuming that the inverse A1 of A exists.

7. The derivative of an operator A () which depends explicitly on a para-meter is defined to be

dA ()

d= lim

0A (+ )A ()

. (2.169)

Show that

d

d(AB) =

dA

dB +A

dB

d. (2.170)


2.14. Problems

8. Show that

d

d

(A1

)= A1dA

dA1 . (2.171)

9. Let |u and |v be two vectors of finite norm. Show thatTr (|u v|) = v |u . (2.172)

10. If A is any linear operator, show that AA is a positive-definite Her-mitian operator whose trace is equal to the sum of the square moduli ofthe matrix elements of A in any arbitrary representation. Deduce thatTr(AA

)= 0 is true if and only if A = 0.

11. Show that ifA andB are two positive-definite observables, thenTr (AB) 0.

12. Show that for any two operators A and L

eLAeL = A+ [L,A]+1

2![L, [L,A]] +

1

3![L, [L, [L,A]]] + . (2.173)

13. Show that ifA andB are two operators satisfying the relation [[A,B] , A] =0 , then the relation

[Am, B] = mAm1 [A,B] (2.174)

holds for all positive integers m .14. Show that

eAeB = eA+Be(1/2)[A,B] , (2.175)

provided that [[A,B] ,A] = 0 and [[A,B] , B] = 0.15. Proof Kondos identity

[A, eH

]= eH

0

eH [H,A] eHd , (2.176)

where A and H are any two operators and is real.16. Show that Tr (XY ) = Tr (Y X).17. Consider the two normalized spin 1/2 states | and |. The operator

A is defined as

A = | | | | . (2.177)Find the eigenvalues of the operator A.

18. A molecule is composed of six identical atoms A1, A2, . . . , A6 whichform a regular hexagon. Consider an electron, which can be localized oneach of the atoms. Let |n be the state in which it is localized on thenth atom (n = 1, 2, , 6). The electron states will be confined to the



space spanned by the states |n, which is assumed to be orthonormal.The Hamiltonian of the system is given by

H = H0 +W . (2.178)

The eigenstates of H0 are the six states |n, with the same eigenvalueE0. The operator W is described by

W |1 = a |2 a |6 ,W |2 = a |3 a |1 ,

...

W |6 = a |1 a |5 .(2.179)

Find the eigenvalues and eigen vectors of H. Clue: Consider a solution ofthe form

|k =6

n=1

eikn |n . (2.180)

2.15 Solutions

1. Let

| = |u+ |v , (2.181)

where C. The requirement | 0 leads to

u |u+ u |v+ v |u+ ||2 v |v 0 . (2.182)

By choosing

= v |uv |v , (2.183)

one has

u |u v |uv |v u |v u |vv |v v |u+

v |uv |v2 v |v 0 , (2.184)

thus

|u |v| u |u

v |v . (2.185)


2.15. Solutions

2. The following holds

(u|+ v|) (|u+ |v) = u |u+ v |v+ 2Re (u |v) u |u+ v |v+ 2 |u |v| .

(2.186)

Thus, using Schwartz inequality one has

(u|+ v|) (|u+ |v) u |u+ v |v+ 2u |u

v |v

=(

u |u+v |v

)2.

(2.187)

3. Since

1 = UU =(1 iF ) (1 + iF ) = 1+ i (F F )+O (2) , (2.188)

one has F = F .4. In general, (| |) = | |, thus clearly the operator A is Hermitian.

Moreover it is positive-definite since for every |u the following holdsu|A |u = u |a a |u = |a |u|2 0 . (2.189)

5. Let

| = |u v|A |uv|A |v |v .

Since A is Hermitian and positive-definite the following holds

0 |A |=

(u| u|A |vv|A |v v|

)A

(|u v|A |uv|A |v |v

)= u|A |u |u|A |v|

2

v|A |v |u|A |v|2v|A |v +

|u|A |v|2v|A |v ,

(2.190)

thus

|u|A |v| u|A |u

v|A |v . (2.191)

Note that this result allows easy proof of the following: Under the sameconditions (namely, A is a Hermitian positive-definite operator) Tr (A) =0 if and only if A = 0.

6. The expansion is given by

(A B)1 = (A (1 A1B))1=(1 A1B)1A1

=(1 + A1B +

(A1B

)2+(A1B

)3+

)A1 .

(2.192)



7. By definition:

d

d(AB) = lim

0A (+ )B (+ )A ()B ()

= lim0

(A (+ )A ())B ()

+ lim0

A (+ ) (B (+ )B ())

=dA

dB +A

dB

d.

(2.193)

8. Taking the derivative of both sides of the identity 1 = AA1 on has

0 =dA

dA1 +A

dA1

d, (2.194)

thus

d

d

(A1

)= A1dA

dA1 . (2.195)

9. Let {|n} be a complete orthonormal basis, namelyn

|n n| = 1 . (2.196)

In this basis

Tr (|u v|) =n

n |u v |n = v|(

n

|n n|)|u = v |u . (2.197)

10. The operator AA is Hermitian since(AA

)= AA, and positive-

definite since the norm of A |u is nonnegative for every |u, thus onehas u|AA |u 0. Moreover, using a complete orthonormal basis {|n}one has

Tr(AA

)=n

n|AA |n

=n,m

n|A |m m|A |n

=n,m

|m|A |n|2 .

(2.198)

11. Let {|b} be a complete orthonormal basis made of eigenvectors of B(i.e., B |b = b |b). Using this basis for evaluating the trace one has

Tr (AB) =bb|AB |b =

b

b0

b|A |b 0

0 . (2.199)


2.15. Solutions

12. Let f (s) = esLAesL, where s is real. Using Taylor expansion one has

f (1) = f (0) +1

1!

df

ds

s=0

+1

2!

d2f

ds2

s=0

+ , (2.200)

thus

eLAeL = A+1

1!

df

ds

s=0

+1

2!

d2f

ds2

s=0

+ , (2.201)

where

df

ds= LesLAesL esLAesLL = [L, f (s)] , (2.202)

d2f

ds2=

[L,df

ds

]= [L, [L, f (s)]] , (2.203)

therefore

eLAeL = A+ [L,A]+1

2![L, [L,A]] +

1

3![L, [L, [L,A]]] + . (2.204)

13. The identity clearly holds for the case m = 1. Moreover, assuming itholds for m, namely assuming that

[Am, B] = mAm1 [A,B] , (2.205)

one has[Am+1,B

]= A [Am, B] + [A,B]Am

= mAm [A,B] + [A,B]Am .

(2.206)

It is easy to show that if [[A,B] , A] = 0 then [[A,B] , Am] = 0, thus oneconcludes that[

Am+1,B]= (m+ 1)Am [A,B] . (2.207)

14. Define the function f (s) = esAesB, where s is real. The following holds

df

ds= AesAesB + esABesB

=(A+ esABesA

)esAesB

Using Eq. (2.174) one has



esAB =m=0

(sA)m

m!B

=m=0

sm (BAm + [Am, B])

m!

=m=0

sm(BAm +mAm1 [A,B]

)m!

= BesA + sm=1

(sA)m1

(m 1)! [A,B]

= BesA + sesA [A,B] ,

(2.208)

thus

df

ds= AesAesB +BesAesB + sesA [A,B] esB

= (A+B + [A,B] s) f (s) .

(2.209)

The above differential equation can be easily integrated since [[A,B] , A] =0 and [[A,B] , B] = 0. Thus

f (s) = e(A+B)se[A,B]s2

2 . (2.210)

For s = 1 one gets

eAeB = eA+Be(1/2)[A,B] . (2.211)

15. Define

f () [A, eH] , (2.212)g () eH

0

eH [H,A] eHd . (2.213)

Clearly, f (0) = g (0) = 0 . Moreover, the following holds

df

d= AHeH +HeHA = Hf + [H,A] eH , (2.214)

dg

d= Hg + [H,A] eH , (2.215)

namely, both functions satisfy the same differential equation. Thereforef = g.

16. Using a complete orthonormal basis {|n} one has


2.15. Solutions

Tr (XY ) =n

n|XY |n

=n,m

n|X |m m|Y |n

=n,m

m|Y |n n|X |m

=m

m|YX |m

= Tr (YX) .

(2.216)

Note that using this result it is easy to show that Tr (U+XU) = Tr (X), provided that U is a unitary operator.

17. Clearly A is Hermitian, namely A = A, thus the two eigenvalues 1and 2 are expected to be real. Since the trace of an operator is basisindependent, the following must hold

Tr (A) = 1 + 2 , (2.217)

and

Tr(A2)= 21 +

22 . (2.218)

On the other hand, with the help of Eq. (2.172) one finds that

Tr (A) = Tr (| |)Tr (| |) = 0 , (2.219)and

Tr(A2)= Tr (| | |) + Tr (| | |)Tr (| | |)Tr (| | |)= 2 |Tr (| |) |Tr (| |)= 2

(1 | ||2

),

(2.220)

thus

= 1 | ||2 . (2.221)

Alternatively, this problem can also be solved as follows. In general, thestate | can be decomposed into a parallel to and a perpendicular to |terms, namely

| = a |+ c | , (2.222)where a, c C, the vector | is orthogonal to |, namely | = 0, andin addition | is assumed to be normalized, namely | = 1. Since |



is normalized one has |a|2 + |c|2 = 1. The matrix representation of A inthe orthonormal basis {| , |} is given by

A=

( |A | |A ||A | |A |

)=

( |c|2 acac |c|2

) A . (2.223)

Thus,

Tr(A)= 0 , (2.224)

and

Det(A)= |c|2

(|c|2 + |a|2

)=

(1 | ||2

), (2.225)

therefore the eigenvalues are

= 1 | ||2 . (2.226)

18. Following the clue, we seek a solution to the eigenvalue equation

H|k = Em |k , (2.227)where

|k =6

n=1

eikn |n . (2.228)

thus

H|k = E0 |k a6

n=1

eikn((n1)+ (n+1)) = E |k , (2.229)

where n is the modulus of n divided by 6 (e.g., 1 = 1, 0 = 6, 7 = 1).A solution is obtained if

e6ik = 1 , (2.230)

or

km =m

3, (2.231)

where m = 1, 2, , 6. The corresponding eigen vectors are denoted as

|km =6

n=1

eikmn |n , (2.232)

and the following holds


3. The Position and Momentum Observables

Consider a point particle moving in a 3 dimensional space. We first treatthe system classically. The position of the particle is described using theCartesian coordinates qx, qy and qz. Let

pj =Lqj

(3.1)

be the canonically conjugate variable to the coordinate qj , where j {x, y, z}and where L is the Lagrangian. As we have seen in exercise 4 of set 1, thefollowing Poissons brackets relations hold

{qj , qk} = 0 , (3.2){pj , pk} = 0 , (3.3){qj , pk} = jk . (3.4)In quantum mechanics, each of the 6 variables qx, qy, qz, px, py and pz is

represented by an Hermitian operator, namely by an observable. It is postu-lated that the commutation relations between each pair of these observablesis related to the corresponding Poissons brackets according to the rule

{, } 1i[, ] . (3.5)

Namely the following is postulated to hold

[qj , qk] = 0 , (3.6)

[pj , pk] = 0 , (3.7)

[qj , pk] = ijk . (3.8)

Note that here we use the same notation for a classical variable and itsquantum observable counterpart. In this chapter we will derive some resultsthat are solely based on Eqs. (3.6), (3.7) and (3.8).

3.1 The One Dimensional Case

In this section, which deals with the relatively simple case of a one dimen-sional motion of a point particle, we employ the less cumbersome notation

Chapter 3. The Position and Momentum Observables

x and p for the observables qx and px. The commutation relation betweenthese operators is given by [see Eq. (3.8)]

[x, p] = i . (3.9)

The uncertainty principle (2.154) employed for x and p yields(x)

2(px)

2

2

4. (3.10)

3.1.1 Position Representation

Let x be an eigenvalue of the observable x, and let |x be the correspondingeigenvector, namely

x |x = x |x . (3.11)

Note that x R since x is Hermitian. As we will see below transformationbetween different eigenvectors |x can be performed using the translationoperator J (x).

Definition 3.1.1. The translation operator is given by

J (x) = exp

( ixp

), (3.12)

where x R.Recall that in general the meaning of a function of an operator can be

understood in terms of the Taylor expansion of the function, that is, for thepresent case

J (x) =n=0

1

n!

( ixp

)n. (3.13)

It is easy to show that J (x) is unitary

J (x) = J (x) = J1 (x) . (3.14)

Moreover, the following composition property holds

J (x1)J (x2) = J (x1 +x2) . (3.15)

Theorem 3.1.1. Let x be an eigenvalue of the observable x, and let |x bethe corresponding eigenvector. Then the ket-vector J (x) |x is a normalizedeigenvector of x with an eigenvalue x +x.


3.1. The One Dimensional Case

Proof. With the help of Eq. (3.77), which is given by

[x,B (p)] = idB

dp, (3.16)

and which is proven in exercise 1 of set 3, one finds that

[x, J (x)] = ixi

J (x) . (3.17)

Using this result one has

xJ (x) |x = ([x, J (x)] + J (x)x) |x = (x +x)J (x) |x , (3.18)thus the ket-vector J (x) |x is an eigenvector of x with an eigenvalue x +x. Moreover, J (x) |x is normalized since J is unitary.

In view of the above theorem we will in what follows employ the notation

J (x) |x = |x +x . (3.19)An important consequence of the last result is that the spectrum of eigenval-ues of the operator x is continuous and contains all real numbers. This pointwill be further discussed below.

The position wavefunction (x) of a state vector | is defined as:

(x) = x | . (3.20)

Given the wavefunction (x) of state vector |, what is the wavefunction

of the state O |, where O is an operator? We will answer this question belowfor some cases:

1. The operator O = x. In this case

x|x | = x x | = x (x) , (3.21)namely, the desired wavefunction is obtained by multiplying (x

) byx.

2. The operator O is a function A (x) of the operator x. Let

A (x) =n

anxn . (3.22)

be the Taylor expansion of A (x). Exploiting the fact that x is Hermitianone finds that

x|A (x) | =n

anx|xn xnx|

| =n

anxn x | = A (x) (x) .

(3.23)


Chapter 3. The Position and Momentum Observables

3. The operator O = J (x). In this case

x|J (x) | = x|J (x) | = x x | = (x x) .(3.24)

4. The operator O = p. In view of Eq. (3.12), the following holds

J (x) = exp(ipx

)= 1 +

ix

p+O((x)

2), (3.25)

thus

x|J (x) | = (x) +ixx| p |+O

((x)

2). (3.26)

On the other hand, according to Eq. (3.24) also the following holds

x|J (x) | = (x +x) . (3.27)Equating the above two expressions for x|J (x) | yields

x| p | = i (x +x) (x)

x+O (x) . (3.28)

Thus, in the limit x 0 one has

x| p | = iddx

. (3.29)

To mathematically understand the last result, consider the differentialoperator

J (x) = exp(x

d

dx

)= 1 +x

d

dx+1

2!

(x

d

dx

)2+ .

(3.30)

In view of the Taylor expansion of an arbitrary function f (x)

f (x0 +x) = f (x0) +xdf

dx+(x)

2

2!

d2f

dx2+

= exp

(x

d

dx

)f

x=x0

= J (x) fx=x0

,

(3.31)

one can argue that the operator J (x) generates translation.


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Eyal Buks - Quantum Mechanics - Lecture Notes