逐字来源、作者主张与候选译文分别保留。工程核验不批准 OCR、译文或数学正确性。
路径积分 · 已登记discussion
Author-named discussion; machine draft source identity and human-review status are preserved.
条件与约定：原有范围、语言、来源类别及审核状态全部保留；本次只确认阅读身份对应，不提升证据资格。
符号：见原文；不跨作者统一符号。
逐字作者来源与版本
然而，还有一种截然不同的方法将经典理论与量子理论联系起来，它完全绕过了哈密顿算子形式，仅使用拉格朗日量。这就是路径积分形式，它基于一种计算时间演化算符的矩阵元的方法
$$
\langle q _ {T} | e ^ {- \frac {i}{\hbar} H T} | q _ {0} \rangle
$$
在位置本征态基下，用对路径空间的积分表示 $_ \mathrm { g o }$ 在时间 $T$ 内从位置 $q _ { 0 }$ 到位置 $q _ { T }$（这里我们只处理 $d = 1 { \mathrm { ~ c a s e ) } }$。这里 $\left| q _ { 0 } \right.$ 是 $Q$ 的一个本征态，特征值为 $q _ { 0 }$（在位置空间表示中为位于 $q _ { 0 }$ 的 delta 函数），而 $\left| q _ { T } \right.$ 具有 $Q$ 特征值 $q _ { T }$。该矩阵元有一个物理解释：它是粒子在 $t = 0$ 时从 $q _ { 0 }$ 出发、在时间 $T ,$ 时具有位置 $q _ { T }$ 的振幅，其范数平方给出在位置 $q _ { T }$ 观测到该粒子的概率密度。它也是核函数，使得人们能够通过计算来确定任意时间 $t = T$ 的波函数 $\psi ( q , T )$，以其在 $t = 0$ 的初值为依据。
$$
\psi (q _ {T}, T) = \int_ {- \infty} ^ {\infty} \langle q _ {T} | e ^ {- \frac {i}{\hbar} H T} | q _ {0} \rangle \psi (q _ {0}, 0) d q _ {0}
$$
路径积分 · 已登记discussion
Author-named discussion; machine draft source identity and human-review status are preserved.
条件与约定：原有范围、语言、来源类别及审核状态全部保留；本次只确认阅读身份对应，不提升证据资格。
符号：见原文；不跨作者统一符号。
逐字作者来源与版本
There is however a very diferent approach to relating classical and quantum theories, which completely bypasses the Hamiltonian formalism, just using the Lagrangian. This is the path integral formalism, which is based upon a method for calculating matrix elements of the time evolution operator
$$
\langle q _ {T} | e ^ {- \frac {i}{\hbar} H T} | q _ {0} \rangle
$$
in the position eigenstate basis in terms of an integral over the space of paths that $_ \mathrm { g o }$ from position $q _ { 0 }$ to position $q _ { T }$ in time $T$ (we will here only treat the $d = 1 { \mathrm { ~ c a s e ) } }$ . Here $\left| q _ { 0 } \right.$ is an eigenstate of $Q$ with eigenvalue $q _ { 0 }$ (a delta-function at $q _ { 0 }$ in the position space representation), and $\left| q _ { T } \right.$ has $Q$ eigenvalue $q _ { T }$ . This matrix element has a physical interpretation as the amplitude for a particle starting at $q _ { 0 }$ at $t = 0$ to have position $q _ { T }$ at time $T ,$ with its norm-squared giving the probability density for observing the particle at position $q _ { T }$ . It is also the kernel function that allows one to determine the wavefunction $\psi ( q , T )$ at any time $t = T$ in terms of its initial value at $t = 0$ , by calculating
$$
\psi (q _ {T}, T) = \int_ {- \infty} ^ {\infty} \langle q _ {T} | e ^ {- \frac {i}{\hbar} H T} | q _ {0} \rangle \psi (q _ {0}, 0) d q _ {0}
$$
Legendre变换 · 已登记definition
Author-named definition; machine draft source identity and human-review status are preserved.
条件与约定：原有范围、语言、来源类别及审核状态全部保留；本次只确认阅读身份对应，不提升证据资格。
符号：见原文；不跨作者统一符号。
逐字作者来源与版本
该变换
$$
\left(q _ {j}, \dot {q} _ {k}\right)\rightarrow \left(q _ {j}, p _ {k} = \frac {\partial L}{\partial \dot {q} _ {k}}\right)
$$
位置-速度空间与相空间之间的变换称为勒让德变换，在良好情形下（例如当 $L$ 对所有速度都是二次的时），它是一个同构。不过一般而言，这并不是同构，勒让德变换常常将位置-速度空间映到相空间的一个较低维子空间。这类情形并不罕见，并且即使作为经典力学系统，也需要复杂得多的形式体系（这一主题称为“约束哈密顿动力学”）。我们将在第 46 章研究的一个重要例子是自由电磁场，其运动方程为麦克斯韦方程。在这种情况下，位形空间坐标是矢势的分量 $\left( A _ { 0 } , A _ { 1 } , A _ { 2 } , A _ { 3 } \right)$，问题之所以出现，是因为拉格朗日量不依赖于 $\dot { A } _ { 0 }$
稳相近似 · 已登记discussion
Author-named discussion; machine draft source identity and human-review status are preserved.
条件与约定：原有范围、语言、来源类别及审核状态全部保留；本次只确认阅读身份对应，不提升证据资格。
符号：见原文；不跨作者统一符号。
逐字作者来源与版本
这个表达式特别吸引人的一个方面在于，它给出了在经典极限 $\hbar 0$ 下经典行为如何出现的简单理解。振荡积分的稳相近似方法指出，对于在 $x = x _ { c } \ ( { \mathrm { i . e . , } } \ f ^ { \prime } ( x _ { c } ) = 0 )$ 处只有一个临界点的函数 $f$，以及小参数 $ϵ$，有
$$
\frac {1}{\sqrt {i 2 \pi \epsilon}} \int_ {- \infty} ^ {+ \infty} d x e ^ {i f / \epsilon} = \frac {1}{\sqrt {f ^ {\prime \prime} (c)}} e ^ {i f (x _ {c}) / \epsilon} (1 + O (\epsilon))
$$
Legendre变换 · 已登记definition
Author-named definition; machine draft source identity and human-review status are preserved.
条件与约定：原有范围、语言、来源类别及审核状态全部保留；本次只确认阅读身份对应，不提升证据资格。
符号：见原文；不跨作者统一符号。
逐字作者来源与版本
The transformation
$$
\left(q _ {j}, \dot {q} _ {k}\right)\rightarrow \left(q _ {j}, p _ {k} = \frac {\partial L}{\partial \dot {q} _ {k}}\right)
$$
between position-velocity and phase space is known as the Legendre transform, and in good cases (for instance when $L$ is quadratic in all the velocities) it is an isomorphism. In general though, this is not an isomorphism, with the Legendre transform often taking position-velocity space to a lower dimensional subspace of phase space. Such cases are not unusual and require a much more complicated formalism, even as classical mechanical systems (this subject is known as “constrained Hamiltonian dynamics”). One important example we will study in chapter 46 is that of the free electromagnetic field, with equations of motion the Maxwell equations. In that case the configuration space coordinates are the components $\left( A _ { 0 } , A _ { 1 } , A _ { 2 } , A _ { 3 } \right)$ of the vector potential, with the problem arising because the Lagrangian does not depend on $\dot { A } _ { 0 }$
稳相近似 · 已登记discussion
Author-named discussion; machine draft source identity and human-review status are preserved.
条件与约定：原有范围、语言、来源类别及审核状态全部保留；本次只确认阅读身份对应，不提升证据资格。
符号：见原文；不跨作者统一符号。
逐字作者来源与版本
An especially attractive aspect of this expression is that it provides a simple understanding of how classical behavior emerges in the classical limit as $\hbar 0$ The stationary phase approximation method for oscillatory integrals says that, for a function $f$ with a single critical point at $x = x _ { c } \ ( { \mathrm { i . e . , } } \ f ^ { \prime } ( x _ { c } ) = 0 )$ and for a small parameter $ϵ$, one has
$$
\frac {1}{\sqrt {i 2 \pi \epsilon}} \int_ {- \infty} ^ {+ \infty} d x e ^ {i f / \epsilon} = \frac {1}{\sqrt {f ^ {\prime \prime} (c)}} e ^ {i f (x _ {c}) / \epsilon} (1 + O (\epsilon))
$$