Carl Bender Mathematical Physics Lecture1 Note

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1

微扰求解 Hard Problem 的步骤

  1. 使某一项成为微扰 \(\epsilon\)
  2. 将无微扰的解加上修正后代回方程, 逐项对应求解
  3. 将所有的项加进来.
  1. Convert the original problem into a perturbation problem by introducing the small parameter \(\epsilon\) .
  2. Assume an expression for the answer in the form of a perturbation series and compute the coefficients of that series.
  3. Recover the answer to the original problem by summing the perturbation series for the appropriate value of \(\epsilon\) .

Symbols

  • \(\sim\) : 'is asympotic to'
\begin{align*} f(x) \sim g(x) \quad (x \to x_0) \end{align*}

means

\begin{align*} \lim_{x\to x_0} \frac{f(x)}{g(x)} = 1 \end{align*}
  • \(\ll\) : 'is negligible compared with'
\begin{align*} f(x) \ll g(x) \quad (x \to x_0) \end{align*}

means

\begin{align*} \lim_{x\to x_0} \frac{f(x)}{g(x)} = 0 \end{align*}

Method of Dominant Balance

求方程

\begin{align*} \epsilon x^5 +x = 1 \end{align*}

的解在 \(\epsilon \to 0\) 时, 行为.

  • Neglect 1
\begin{align*} \epsilon x^5 \sim -x \end{align*}

得到

\begin{align*} x \sim \frac{(-1)^{1/4}}{\epsilon^{1/4}} \end{align*}

是复平面上四个在无穷远的解. \(1\) 相比无穷大, 可以忽略, 不矛盾.

  • Neglect \(x\)
\begin{align*} \epsilon x^5 \sim 1 \end{align*}

得到

\begin{align*} x \sim \frac{1^{1/5}}{\epsilon^{1/5}} \end{align*}

这里无穷大的 \(x\) 相比于 \(1\) 是不可忽略的. 矛盾

  • Neglect \(\epsilon x^5\)
\begin{align*} x \sim 1 \end{align*}

那么

\begin{align*} \epsilon x^5 \sim \epsilon\cdot 1 \ll 1 \end{align*}

可以忽略.