Chapter 4 Path Integral,4.1 Classical action and the amplitude in Quantum Mechanics,Introduction: how to quantize? Wave mechanics h Schrdinger equ. Matrix mechanics h commutator Classical Poisson bracket Q. P. B. Path integral h wave function,4.1 Classical action and the amplitude in Quantum Mechanics,Basic idea Infinite orbits Different orbits have different probabilities,4.1 Classical action and the amplitude in Quantum Mechanics,A particle starting from a certain initial state may reach the final state through different possible orbits with different probabilities,4.1 Classical action and the amplitude in Quantum Mechanics,Classical action,4.1 Classical action and the amplitude in Quantum Mechanics,4.1 Classical action and the amplitude in Quantum Mechanics,4.1 Classical action and the amplitude in Quantum Mechanics,4.1 Classical action and the amplitude in Quantum Mechanics,Free particle,4.1 Classical action and the amplitude in Quantum Mechanics,4.1 Classical action and the amplitude in Quantum Mechanics,Linear oscillator,4.1 Classical action and the amplitude in Quantum Mechanics,4.1 Classical action and the amplitude in Quantum Mechanics,4.1 Classical action and the amplitude in Quantum Mechanics,4.1 Classical action and the amplitude in Quantum Mechanics,Amplitude in quantum mechanics All paths, not only just one path from a to b, have contributions The contributions of all paths to probability amplitude are the same in module, but different in phases The contribution of the phase from each path is proportional to S/h, where S is the action of the corresponding path,4.1 Classical action and the amplitude in Quantum Mechanics,In summary: the quantization scheme of the path integral supposes that the probability P(a, b) of the transition is,4.1 Classical action and the amplitude in Quantum Mechanics,4.1 Classical action and the amplitude in Quantum Mechanics,h appears as a part of the phase factor Q.M. C.M while h 0,4.1 Classical action and the amplitude in Quantum Mechanics,Classical limit: S/h 1 Quickly oscillate,4.1 Classical action and the amplitude in Quantum Mechanics,S depends on xa, xb considerably,4.2 Path integral,How to calculate K(b, a),4.2 Path integral,Key: the variable in the integration is a function This is a functional integral,4.2 Path integral,4.2 Path integral,4.2 Path integral,4.2 Path integral,4.2 Path integral,4.2 Path integral,The functional integration of two adjacent events,4.2 Path integral,4.2 Path integral,4.2 Path integral,4.2 Path integral,4.2 Path integral,Free particles Additional normalization factor,4.2 Path integral,4.2 Path integral,4.2 Path integral,4.2 Path integral,4.2 Path integral,4.2 Path integral,de Broglie relation,4.2 Path integral,4.2 Path integral,4.2 Path integral,4.2 Path integral,4.2 Path integral,Normalization factor,4.2 Path integral,4.2 Path integral,4.3 Gauss integration,A type of functional integration which can easily be calculated,4.3 Gauss integration,4.3 Gauss integration,4.3 Gauss integration,4.3 Gauss integration,Conclusion: The Gauss integration only depends on the second homogeneous function of y and derivative of y,4.3 Gauss integration,Normalization factor of the linear oscillator,4.3 Gauss integration,4.3 Gauss integration,4.3 Gauss integration,Forced oscillator situation,4.3 Gauss integration,4.3 Gauss integration,Any potential,4.3 Gauss integration,4.4 Path integral and the Schrdinger equation,Path integral Schrdinger equation Path integral wave mechanics matrix mechanics,4.4 Path integral and the Schrdinger equation,1D free particle,4.4 Path integral and the Schrdinger equation,4.4 Path integral and the Schrdinger equation,4.4 Path integral and the Schrdinger equation,4.4 Path integral and the Schrdinger equation,With effective potential,4.4 Path integral and the Schrdinger equation,4.4 Path integral and the Schrdinger equation,4.4 Path integral and the Schrdinger equation,4.4 Path integral and the Schrdinger equation,4.4 Path integral and the Schrdinger equation,4.4 Path integral and the Schrdinger equation,3D Schrdinger equation,4.4 Path integral and the Schrdinger equation,4.4 Path integral and the Schrdinger equation,4.4 Path integral and the Schrdinger equation,4.4 Path integral and the Schrdinger equation,4.4 Path integral and the Schrdinger equation,4.5 The canonical form of the path integral,4.5 The canonical form of the path integral,4.5 The canonical form of the path integral,4.5 The canonical form of the path integral,4.5 The canonical form of the path integral,4.5 The canonical form of the path integral,4.5 The canonical form of the path integral,4.5 The canonical form of the path integral,4.5 The canonical form of the path integral,4.5 The canonical form of the path integral,Conclusion: canonical form Lagrange form,