复旦量子力学讲义qmchapter

1、Chapter 4Path Integral Classical action and the amplitude in Quantum MechanicsIntroduction: how to quantize?Wave mechanics h Schrdinger equ.Matrix mechanics h commutator Classical Poisson bracket Q. P. B.Path integral h wave function Classical action and the amplitude in Quantum MechanicsBasic ideaI

2、nfinite orbitsDifferent orbits have different probabilities 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 probabilities Classical action and the amplitude in Quantum M

3、echanicsClassical action Classical action and the amplitude in Quantum Mechanics Classical action and the amplitude in Quantum Mechanics Classical action and the amplitude in Quantum Mechanics Classical action and the amplitude in Quantum MechanicsFree particle Classical action and the amplitude in

4、Quantum Mechanics Classical action and the amplitude in Quantum MechanicsLinear oscillator Classical action and the amplitude in Quantum Mechanics Classical action and the amplitude in Quantum Mechanics Classical action and the amplitude in Quantum Mechanics Classical action and the amplitude in Qua

5、ntum MechanicsAmplitude 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 acti

6、on of the corresponding path Classical action and the amplitude in Quantum MechanicsIn summary: the quantization scheme of the path integral supposes that the probability P(a, b) of the transition is Classical action and the amplitude in Quantum Mechanics Classical action and the amplitude in Quantu

7、m Mechanicsh appears as a part of the phase factorQ.M. C.M while h 0 Classical action and the amplitude in Quantum MechanicsClassical limit: S/h 1Quickly oscillate Classical action and the amplitude in Quantum MechanicsS depends on xa, xb considerably Path integralHow to calculate K(b, a) Path integ

8、ralKey: the variable in the integration is a function This is a functional integral Path integral Path integral Path integral Path integral Path integral Path integralThe functional integration of two adjacent events Path integral Path integral Path integral Path integral Path integralFree particles

9、Additional normalization factor Path integral Path integral Path integral Path integral Path integral Path integralde Broglie relation Path integral Path integral Path integral Path integral Path integralNormalization factor Path integral Path integral Gauss integrationA type of functional integrati

10、on which can easily be calculated Gauss integration Gauss integration Gauss integration Gauss integrationConclusion: The Gauss integration only depends on the second homogeneous function of y and derivative of y Gauss integrationNormalization factor of the linear oscillator Gauss integration Gauss i

11、ntegration Gauss integrationForced oscillator situation Gauss integration Gauss integrationAny potential Gauss integration Path integral and the Schrdinger equationPath integral Schrdinger equationPath integral wave mechanics matrix mechanics Path integral and the Schrdinger equation1D free particle

12、 Path integral and the Schrdinger equation Path integral and the Schrdinger equation Path integral and the Schrdinger equation Path integral and the Schrdinger equationWith effective potential Path integral and the Schrdinger equation Path integral and the Schrdinger equation Path integral and the S

13、chrdinger equation Path integral and the Schrdinger equation Path integral and the Schrdinger equation Path integral and the Schrdinger equation3D Schrdinger equation Path integral and the Schrdinger equation Path integral and the Schrdinger equation Path integral and the Schrdinger equation Path integral and the Schrdinger equation Path integral and the Schrdinger equation The canonical form of the path integral The canonical form of the path integral The canonical form of the path integral The canonical form of t