xii Contents 82 Path integrals 11T 8.2. Derivation 822 Mode expansion 8.2.3 Feynman graphs 8.2.4 Back to ordinary (Minkowski)time 8.2.5 Tunnel effects and instantons 198 Further reading 201 Numerical analyses 202 II Approximation methods 203 9 Perturbation theory 9.1 Time-independent perturbations 011n generate levels 212 9.1.2 The Stark effect on the=2level of the hydroger 9.13 Dipoleinteractionsandpolarizabhlty 9.2 Quantum transitions 92.1 Perturbation lasting for a finite interval 9.2.2 Periodic perturbation 9.2.3 Transitions in a discrete spectrum 223 9.2.4 Resonant oscillation between two levels 225 9.3 Transitions in the continuum 226 12 9.3.1 State density 228 228 9.5 omagnetic transitions The dipole oximation diation (or stimulated)emission Spontaneous emission 9.6 The Einstein coefficients Guide to the Supplements 湖新源幼4422 Problems Numerical analyses 244 10 Variational methods 245 10.1 The variational principle 10.1.1 Lower limits 10.1.2 Truncated Hilbert space 249 10.2 Simple applications 10.21Te 1022H oscilator elementary variational calculatior 10.2.3Ths virial 10.3 The ground state of the helium Guide to the Supplements Problems Numerical analyses 262