Contents Interaction picture 59 3.4 Superposition of time-dependent states and energy-time 63 35 Probability conservation 3.7 Ehrenfest's theorem 68 3.8 Problems 70 Free particles 73 4.1 Free particle in one dimension 73 4.2 Normalization 75 4.3 Momentum eigenfunctions and Fourier transforms Minimum uncertainty wave packet 13 45 Group velocity of a superposition of plane waves 3 4.6 Three dimensions-Cartesian coordinates 84 Three dimensions-spherical coordinates The radial wave equation 9 4.9 Properties of Yim(,) 92 4.10 Angular momentum 9 4.11 Determining L2from th ngular variables Commutator Li,L and L2 98 4.13 Ladder operators 100 4.14 Problems 102 5 Particles with spin V 103 5.1 Spin⅓system 103 5.2 Pauli matrices 104 The spin⅓eigenstates Matrix representation of ox and oy 8 5.5 Eigenstates of o and o. 108 5.6 eigenstates of spin in an arbitrary direction 109 5.7 e important relations for 110 Arbitrary 2 x 2 matrices in terms of Pauli matrices 5.9 Projection operator for spin systems 112 5.10 Density matrix for spin states and the ensemble average 114 Compete wavefunctio 116 Pauli exclusion principle and Fermi energy 116 5.13 Problems 118 6 Gauge invariance,angular momentum,and spin 6 Gauge invariance 18 6.2 Quantum mechanics 121 Canonical and kinematic momenta 123 64 Probability conservation 124viii Contents 3.3 Interaction picture 59 3.4 Superposition of time-dependent states and energy–time uncertainty relation 63 3.5 Time dependence of the density operator 66 3.6 Probability conservation 67 3.7 Ehrenfest’s theorem 68 3.8 Problems 70 4 Free particles 73 4.1 Free particle in one dimension 73 4.2 Normalization 75 4.3 Momentum eigenfunctions and Fourier transforms 78 4.4 Minimum uncertainty wave packet 79 4.5 Group velocity of a superposition of plane waves 83 4.6 Three dimensions – Cartesian coordinates 84 4.7 Three dimensions – spherical coordinates 87 4.8 The radial wave equation 91 4.9 Properties of Ylm(θ, φ) 92 4.10 Angular momentum 94 4.11 Determining L2 from the angular variables 97 4.12 Commutator Li, Lj and L2, Lj 98 4.13 Ladder operators 100 4.14 Problems 102 5 Particles with spin ½ 103 5.1 Spin ½ system 103 5.2 Pauli matrices 104 5.3 The spin ½ eigenstates 105 5.4 Matrix representation of σx and σy 106 5.5 Eigenstates of σx and σy 108 5.6 Eigenstates of spin in an arbitrary direction 109 5.7 Some important relations for σi 110 5.8 Arbitrary 2 × 2 matrices in terms of Pauli matrices 111 5.9 Projection operator for spin ½ systems 112 5.10 Density matrix for spin ½ states and the ensemble average 114 5.11 Complete wavefunction 116 5.12 Pauli exclusion principle and Fermi energy 116 5.13 Problems 118 6 Gauge invariance, angular momentum, and spin 120 6.1 Gauge invariance 120 6.2 Quantum mechanics 121 6.3 Canonical and kinematic momenta 123 6.4 Probability conservation 124