复旦量子力学讲义qmchapter

Chapter 4Path Integral4.1 Classical action and the amplitude in Quantum MechanicsIntroduction: how to quantize?lWave mechanics h Schrdinger equ.lMatrix mechanics h commutator Classical Poisson bracket Q. P. B.lPath integral h wave function4.1 Classical action and the amplitude in Quantum MechanicsBasic idealInfinite orbitslDifferent orbits have different probabilities4.1 Classical action and the amplitude in Quantum MechanicsA particle starting from a certain initial state may reach the final state through different possible orbits with different probabilities4.1 Classical action and the amplitude in Quantum MechanicsClassical action4.1 Classical action and the amplitude in Quantum Mechanics4.1 Classical action and the amplitude in Quantum Mechanics4.1 Classical action and the amplitude in Quantum Mechanics4.1 Classical action and the amplitude in Quantum MechanicslFree particle4.1 Classical action and the amplitude in Quantum Mechanics4.1 Classical action and the amplitude in Quantum MechanicslLinear oscillator4.1 Classical action and the amplitude in Quantum Mechanics4.1 Classical action and the amplitude in Quantum Mechanics4.1 Classical action and the amplitude in Quantum Mechanics4.1 Classical action and the amplitude in Quantum MechanicslAmplitude in quantum mechanicsAll paths, not only just one path from a to b, have contributionsThe contributions of all paths to probability amplitude are the same in module, but different in phasesThe contribution of the phase from each path is proportional to S/h, where S is the action of the corresponding path4.1 Classical action and the amplitude in Quantum MechanicslIn summary: the quantization scheme of the path integral supposes that the probability P(a, b) of the transition is4.1 Classical action and the amplitude in Quantum Mechanics4.1 Classical action and the amplitude in Quantum Mechanicslh appears as a part of the phase factorlQ.M. C.M while h 04.1 Classical action and the amplitude in Quantum MechanicsClassical limit: S/h 1Quickly oscillate4.1 Classical action and the amplitude in Quantum MechanicslS depends on xa, xb considerably4.2 Path integralHow to calculate K(b, a)4.2 Path integralKey: the variable in the integration is a function This is a functional integral4.2 Path integral4.2 Path integral4.2 Path integral4.2 Path integral4.2 Path integral4.2 Path integrallThe functional integration of two adjacent events4.2 Path integral4.2 Path integral4.2 Path integral4.2 Path integral4.2 Path integrallFree particleslAdditional normalization factor4.2 Path integral4.2 Path integral4.2 Path integral4.2 Path integral4.2 Path integral4.2 Path integrallde Broglie relation4.2 Path integral4.2 Path integral4.2 Path integral4.2 Path integral4.2 Path integrallNormalization factor4.2 Path integral4.2 Path integral4.3 Gauss integrationA type of functional integration which can easily be calculated4.3 Gauss integration4.3 Gauss integration4.3 Gauss integration4.3 Gauss integrationConclusion: The Gauss integration only depends on the second homogeneous function of y and derivative of y4.3 Gauss integrationlNormalization factor of the linear oscillator4.3 Gauss integration4.3 Gauss integration4.3 Gauss integrationlForced oscillator situation4.3 Gauss integration4.3 Gauss integrationlAny potential4.3 Gauss integration4.4 Path integral and the Schrdinger equationPath integral Schrdinger equationPath integral wave mechanics matrix mechanics4.4 Path integral and the Schrdinger equation1D free particle4.4 Path integral and the Schrdinger equation4.4 Path integral and the Schrdinger equation4.4 Path integral and the Schrdinger equation4.4 Path integral and the Schrdinger equationWith effective potential4.4 Path integral and the Schrdinger equation4.4 Path integral and the Schrdinger equation4.4 Path integral and the Schrdinger equation4.4 Path integral and the Schrdinger equation4.4 Path integral and the Schrdinger equation4.4 Path integral and the Schrdinger equation3D Schrdinger equation4.4 Path integral and the Schrdinger equation4.4 Path integral and the Schrdinger equation4.4 Path integral and the Schrdinger equation4.4 Path integral and the Schrdinger equation4.4 Path integral and the Schrdinger equation4.5 The canonical form of the path integral4.5 The canonical form of the path integral4.5 The canonical form of the path integral4.5 The canonical form of the path integral4.5 The canonical form of the path integral4.5 The canonical form of the path integral4.5 The canonical form of the path integral4.5 The canonical form of the path integral4.5 The canonical form of the path integral4.5 The canonical form of the path integralConclusion: canonical form Lagrange form