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# 物理代写|量子力学代写Quantum mechanics代考|PHYS2941 Historical and Other Comments on the Hamilton-Jacobi Equation

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## 物理代写|量子力学代写Quantum mechanics代考|Historical and Other Comments on the Hamilton-Jacobi Equation

A version of the Hamilton-Jacobi equation first appeared in Hamilton’s researches in the 1830’s into variational formulations of mechanics and their relation to optics. Hamilton discovered the form of the classical equations of motion that now bears his name, and he realized that their complete solution was equivalent to solving a certain partial differential equation, Eq. (39). These ideas were generalized by Jacobi a few years later, who developed methods for solving Eq. (32), what we now call the time-independent Hamilton-Jacobi equation. Jacobi used these methods to solve some nontrivial problems in classical mechanics.

Renewed interest in the Hamilton-Jacobi equation arose in the period of the old quantum theory (1900-1925), after Sommerfeld’s and Wilson’s analysis of Bohr’s quantization condition, which is expressed in terms of action integrals like (13). The old quantum theory was a collection of rules that were incomplete, logically unclear, and of limited applicability, but which gave excellent agreement with experiment in a number of important cases such as the specific heat of solids and the spectrum of hydrogen, including its fine structure. As a result, the period 1911-1925 saw a heightened interest in the formal structure of classical mechanics, in the hope that it would elucidate the difficulties of the old quantum theory. These efforts were summarized in Born’s book The Mechanics of the Atom, published just about the time that Heisenberg’s and Schrödinger’s (modern) quantum theory emerged. After this, interest in classical mechanics, at least in physics circles, fell to almost zero.
In more recent times, since the advent of computers and the discovery of chaos, classical mechanics has enjoyed another revival, partly stimulated also by developments in mathematics such as the KAM (Kolmogorov-Arnold-Moser) theorem. It is now recognized that the Hamilton-Jacobi equation has no global solutions in the case of chaotic motion, and that this has an impact on the morphology and other features of the quantum wave function.

In the language of classical mechanics, the solution $S$ of the Hamilton-Jacobi equation is the generator of the canonical transformation that trivializes the classical equations of motion. The existence of this transformation requires that the system have a sufficient number of commuting constants of motion (the classical analog of a complete set of commuting observables). The constants of motion are conveniently expressed as functions of the actions, themselves constants of motion that generate periodic (and commuting) flows in phase space. The variables canonically conjugate to the actions are certain angles. Such matters are discussed in advanced courses in classical mechanics.
Because of problems with chaos and other issues, the Hamilton-Jacobi equation and the other equations of WKB theory are harder to solve in the multidimensional case, so the most common applications of WKB theory are in one dimension. We now turn to that case.

## 物理代写|量子力学代写Quantum mechanics代考|One-Dimensional WKB Problems

We consider now the one-dimensional Schrödinger equation,
$$-\frac{\hbar^2}{2 m} \psi^{\prime \prime}(x)+V(x) \psi(x)=E \psi(x)$$
in which we use the one-dimensional WKB ansatz,
$$\psi(x)=A(x) e^{i S(x) / \hbar}$$
The Hamilton-Jacobi equation is the one-dimensional version of Eq. (25),
$$\frac{1}{2 m}\left(\frac{d S}{d x}\right)^2+V(x)=E$$

and the amplitude transport equation is the one-dimensional version of Eq. (29),
$$\frac{d}{d x}\left(A^2 \frac{d S}{d x}\right)=0$$
We solve the Hamilton-Jacobi equation (44) algebraically for $d S / d x$, obtaining
$$\frac{d S}{d x}=p(x)= \pm \sqrt{2 m[E-V(x)]}$$

## 物理代写|量子力学代写Quantum mechanics代考|Historical and Other Comments on the Hamilton-Jacobi Equation

Hamilton-Jacobi 方程的一个版本首先出现在 Hamilton 在 1830 年代对力学的变分公式及其与光学的关系的研究中。汉密尔顿发现了现在以他的名字命名的经典运动方程的形式，并且他意识到它们的完整解等同于求解某个偏微分方程，Eq。(39)。几年后，雅可比推广了这些想法，他开发了求解方程式的方法。(32)，我们现在称之为时间无关的 Hamilton-Jacobi 方程。雅可比用这些方法解决了经典力学中的一些重要问题。

## 物理代写|量子力学代写Quantum mechanics代考|One-Dimensional WKB Problems

$$-\frac{\hbar^2}{2 m} \psi^{\prime \prime}(x)+V(x) \psi(x)=E \psi(x)$$

$$\psi(x)=A(x) e^{i S(x) / \hbar}$$
Hamilton-Jacobi 方程是方程式的一维版本。(25)，
$$\frac{1}{2 m}\left(\frac{d S}{d x}\right)^2+V(x)=E$$

$$\frac{d}{d x}\left(A^2 \frac{d S}{d x}\right)=0$$

$$\frac{d S}{d x}=p(x)= \pm \sqrt{2 m[E-V(x)]}$$

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