6 The mechanics of Quantum Mechanics 169 6.1 Abstract Vector Spaces 169 Matrix Representation of a Vector 6.1.2 Dirac notation for a vector 172 6.1.3 Operators in Quantum Mechanics The Eigenvalue Equation 179 Properties of Hermitian Operators and the Eigenvalue Equation 6.2.2 Properties of Commutators 6.3 The Postulates of Quantum Mechanics Listing of the postulates 6.3.2 Discussion of the postulates Further Consequences of the Postulates 6.4 Relation between the state Vector and the wave function.200 6.5 The Heisenberg Picture 202 6.6 Spreading of wa 6.6.1 Spreading in the Heisenberg Picture Spreading in the Schrodinger Picture 211 6.7 Retrospective 216 References 217 Problems 7 Harmonic Oscillator Solution Using Operator Methods 7.1 The Algebraic Method 219 7.1.1 The Schrodinger Picture 7.1.3 he Heisenberg Picture 7.2 Coherent States of the harmonic oscillator 7.3 Retrospective Reference 236 Problems 237 8 Quantum Mechanics in Three Dimensions-Angular Momentum .. 239 8.1 Commutation Relations 240 8.2 Angular Momentum Ladder Operators Definitions and Commutation relations 8.2.2 Angular Momentum Eigenvalues 8.3 Vector Operators 247 Orbital Angular Momentum Eigenfunctions--Spherical Harmonics 249 8.4.1 e Addition Theorem for Spherical Harmonics. 257 Parity 8.4.3 The Rigid Rotor Another Form of Angular Momentum-SContents xiii 6 The Mechanics of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 169 6.1 Abstract Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 6.1.1 Matrix Representation of a Vector . . . . . . . . . . . . . . 171 6.1.2 Dirac Notation for a Vector . . . . . . . . . . . . . . . . . . . . 172 6.1.3 Operators in Quantum Mechanics . . . . . . . . . . . . . . . 173 6.2 The Eigenvalue Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 6.2.1 Properties of Hermitian Operators and the Eigenvalue Equation . . . . . . . . . . . . . . . . . . 180 6.2.2 Properties of Commutators . . . . . . . . . . . . . . . . . . . . 186 6.3 The Postulates of Quantum Mechanics. . . . . . . . . . . . . . . . . . . . . . 189 6.3.1 Listing of the Postulates . . . . . . . . . . . . . . . . . . . . . . . 189 6.3.2 Discussion of the Postulates . . . . . . . . . . . . . . . . . . . 190 6.3.3 Further Consequences of the Postulates . . . . . . . . . . 198 6.4 Relation Between the State Vector and the Wave Function . . . . . 200 6.5 The Heisenberg Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 6.6 Spreading of Wave Packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 6.6.1 Spreading in the Heisenberg Picture . . . . . . . . . . . . 207 6.6.2 Spreading in the Schr¨odinger Picture . . . . . . . . . . . . 211 6.7 Retrospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 6.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 7 Harmonic Oscillator Solution Using Operator Methods . . . . . . . . . . . . 219 7.1 The Algebraic Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 7.1.1 The Schr¨odinger Picture . . . . . . . . . . . . . . . . . . . . . . 219 7.1.2 Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 7.1.3 The Heisenberg Picture . . . . . . . . . . . . . . . . . . . . . . . 227 7.2 Coherent States of the Harmonic Oscillator . . . . . . . . . . . . . . . . . . 229 7.3 Retrospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 7.4 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 8 Quantum Mechanics in Three Dimensions—Angular Momentum . . . 239 8.1 Commutation Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 8.2 Angular Momentum Ladder Operators . . . . . . . . . . . . . . . . . . . . . . 241 8.2.1 Definitions and Commutation Relations . . . . . . . . . 241 8.2.2 Angular Momentum Eigenvalues . . . . . . . . . . . . . . . 242 8.3 Vector Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 8.4 Orbital Angular Momentum Eigenfunctions—Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 8.4.1 The Addition Theorem for Spherical Harmonics . . 257 8.4.2 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 8.4.3 The Rigid Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 8.5 Another Form of Angular Momentum—Spin . . . . . . . . . . . . . . . . 262