苏汝铿高等量子力学讲义(英文版)Chapter4 Path Integral

§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
§4.1 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

§4.4 Path integral and the Schrödinger equation

1D free particle
§4.4 Path integral and the Schrödinger equation
§4.4 Path integral and the Schrödinger equation

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

Free particle
§4.1 Classical action and the amplitude in Quantum Mechanics
§4.1 Classical action and the amplitude in Quantum Mechanics
§4.4 Path integral and the Schrödinger equation
§4.4 Path integral and the Schrödinger equation

With effective potential
§4.4 Path integral and the Schrödinger equation


§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
§4.4 Path integral and the Schrödinger equation
§4.4 Path integral and the Schrödinger equation
§4.4 Path integral and the Schrödinger equation
§4.4 Path integral and the Schrödinger equation
§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
§4.4 Path integral and the Schrödinger equation

3D Schrödinger equation
§4.4 Path integral and the Schrödinger equation
§4.4 Path integral and the Schrödinger equation
§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

S depewk.baidu.comds 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
§4.4 Path integral and the Schrödinger equation
§4.4 Path integral and the Schrödinger equation
§4.4 Path integral and the Schrödinger equation
§4.5 The canonical form of the path integral
§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
§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
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
§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
§4.1 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
§4.3 Gauss integration
§4.3 Gauss integration

Any potential
§4.3 Gauss integration
§4.4 Path integral and the Schrödinger equation
Path integral Schrödinger equation Path integral wave mechanics matrix mechanics
§4.5 The canonical form of the path integral

Conclusion: canonical form Lagrange form
§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
Chapter 4 Path Integral
§4.1 Classical action and the amplitude in Quantum Mechanics



Introduction: how to quantize? Wave mechanics h Schrödinger equ. Matrix mechanics h commutator Classical Poisson bracket Q. P. B. Path integral h wave function

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
§4.2 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
§4.5 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