11 The semi-classical approximation 265 11.1 The WKB approximation 265 111.1 Connection formulas 268 11.2 The Bohr-Sommerfeld quantization condition 271 11.2.1 Counting the quantum states 273 11.2.2 Potentials defined for >0only 2 the meaning of the mt 0 124 11.2.5 11.2.6 Examples 282 11.3 The tunnel effect 283 11.3.1 The double well 285 11.3.2 The semi-classical treatment of decay processes 2 11.3.3 The Gamow-Siegert theory 11.4 Phase shift 295 Further reading 300 300 0咖 Ill Applications 303 12 Time neral fe tures of time evolution 12.2T -dependent unitary transformations 12.3Ad abatic processes 311 12.3.1 The Landau-Zener transition 313 12.3.2 The impulse approximation 12.3.3 The Berry phase 12.3.4 Examples 318 12.4 Some nontrivial systems 320 12.4.1 A particle within moving walls 320 10A9R nant oscillati 324 1943A 1244Agw他 gasolenoid 12. 331 12.5】 Inverse linear variation of the frequency 335 12.5.2 The Planck distribution inside an oscillating cavity 336 12.5.3 General power-dependent frequencies 338 12.5.4 Exponential dependence 12.5.5 Creation and annihilation operators:coupled os- cillators 340 Guide to the Supplements 341 Problems 34 Numerical analyses 42 13 Metastable states 343