# 路徑積分表述

## 數學方法

### 哈密頓算符在量子力學中的意義

${\displaystyle U(t_{b},t_{a})=e^{-{\frac {i}{\hbar }}(t_{b}-t_{a})H}.}$

${\displaystyle iG(x_{b},t_{b};x_{a},t_{a})\equiv \left\langle x_{b}\right|U(t_{b},t_{a})\left|x_{a}\right\rangle .}$

${\displaystyle U(t_{b},t_{a})=U(t_{b},t)U(t,t_{a}),}$

${\displaystyle iG(x_{b},t_{b};x_{a},t_{a})=\int dx\ iG(x_{b},t_{b};x,t)iG(x,t;x_{a},t_{a})}$

### 時間切片

{\displaystyle {\begin{aligned}\left\langle x_{j}\left|e^{-i{\frac {\Delta }{\hbar }}H({\hat {p}},{\hat {x}})}\right|x_{j-1}\right\rangle &=\int dp_{j}\langle x_{j}|p_{j}\rangle \left\langle p_{j}\left|e^{-i{\frac {\Delta }{\hbar }}H({\hat {p}},{\hat {x}})}\right|x_{j-1}\right\rangle \end{aligned}}}

${\displaystyle e^{-i{\frac {\Delta }{\hbar }}H({\hat {p}},{\hat {x}})}=:e^{-i{\frac {\Delta }{\hbar }}H({\hat {p}},{\hat {x}})}:+O(\Delta ^{2})}$

{\displaystyle {\begin{aligned}\left\langle x_{j}\left|e^{-i{\frac {\Delta }{\hbar }}H({\hat {p}},{\hat {x}})}\right|x_{j-1}\right\rangle &=\int {\frac {dp_{j}}{2\pi \hbar }}e^{i{\frac {p_{j}}{\hbar }}(x_{j}-x_{j-1})}\,e^{-i{\frac {\Delta }{\hbar }}H(p_{j},x_{j-1})}\\&=\int {\frac {dp_{j}}{2\pi \hbar }}e^{i{\frac {\Delta }{\hbar }}\left(p_{j}{\frac {x_{j}-x_{j-1}}{\Delta }}-H(p_{j},x_{j-1})\right)}\\\end{aligned}}}

{\displaystyle {\begin{aligned}iG(x_{b},t_{b};x_{a},t_{a})&=\int dx_{1}\cdots dx_{n-1}\prod _{i=1}^{n-1}dp_{i}\exp \left[{\frac {i}{\hbar }}\sum _{j=1}^{n-1}\Delta \,L\left(t_{j},{\frac {x_{j}+x_{j-1}}{2}},{\frac {x_{j}-x_{j-1}}{\Delta }}\right)\right]\\&=\int {\mathcal {D}}\left[x(t)\right]e^{{\frac {i}{\hbar }}S[x(t)]},\end{aligned}}}

${\displaystyle S=\int L(t,x,{\dot {x}})dt.}$

## 简單例子

### 自由粒子

${\displaystyle S=\int {\frac {{\dot {x}}^{2}}{2}}dt,}$

${\displaystyle G(x-y;T)=\int _{x(0)=x}^{x(T)=y}e^{-\int _{0}^{T}{\frac {{\dot {x}}^{2}}{2}}dt}{\mathcal {D}}x=\int _{x(0)=x}^{x(T)=y}\prod _{t}e^{-{\frac {1}{2}}\left({\frac {(x(t+\epsilon )-x(t)}{\epsilon }}\right)^{2}\epsilon }{\mathcal {D}}x,}$

${\displaystyle G(x-y;T)=G_{\epsilon }*G_{\epsilon }*G_{\epsilon }*\cdots *G_{\epsilon }(x-y).}$

${\displaystyle {\tilde {G}}(p;T)={\tilde {G}}_{\epsilon }(p)^{T/\epsilon }.}$

${\displaystyle {\tilde {G}}_{\epsilon }(p)=e^{-\epsilon {\frac {p^{2}}{2}}},}$

${\displaystyle {\tilde {G}}(p;T)=e^{-T{\frac {p^{2}}{2}}}.}$

${\displaystyle G(x-y;T)\propto e^{-{\frac {(x-y)^{2}}{2T}}}.}$

${\displaystyle \int G(x-y;T)dy=1,}$

${\displaystyle {\frac {d}{dt}}G(x;t)={\frac {\nabla ^{2}}{2}}G.}$

${\displaystyle G(x-y;T)\propto e^{\frac {i(x-y)^{2}}{2T}}.}$

${\displaystyle {\frac {d}{dt}}G(x;t)={\frac {i\nabla ^{2}}{2}}G.}$

${\displaystyle \varphi _{t}(x)=\int \varphi _{0}(y)G(x-y;t)dy,}$

${\displaystyle G}$一樣服從薛定諤方程式：

${\displaystyle i{\frac {d}{dt}}\varphi _{t}=-{\frac {\nabla ^{2}}{2}}\varphi _{t}(x).}$

## 量子场论

${\displaystyle Z=\int D\phi \ \exp(iS(\phi ))}$

## 参考资料

1. ^ Chaichian, Masud; Demichev, Andrei Pavlovich. Introduction. Path Integrals in Physics Volume 1: Stochastic Process & Quantum Mechanics. Taylor & Francis. 2001: 1ff. [2016-10-21]. ISBN 0-7503-0801-X. （原始内容存档于2019-05-02）.
2. ^ Dirac, Paul A. M. The Lagrangian in Quantum Mechanics (PDF). Physikalische Zeitschrift der Sowjetunion. 1933, 3: 64–72 [2016-10-21]. （原始内容存档 (PDF)于2017-01-14）.
3. ^ Van Vleck, John H. The correspondence principle in the statistical interpretation of quantum mechanics. Proceedings of the National Academy of Sciences of the United States of America. 1928, 14 (2): 178–188. Bibcode:1928PNAS...14..178V. . PMID 16577107. doi:10.1073/pnas.14.2.178.