## Information

• 原出处: https://www.perimeterinstitute.ca/video-library/collection/11/12-psi-mathematical-physics
• Bilibili: https://www.bilibili.com/video/BV1w4411q7x6?fromsearch&seid7852838902448285010
• Book: Carl M. Bender, Steven A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I Asympotic Methods and Perturbation Theory, 1999

## Keywords

The Schrodinger equation. Riccati equation. Initial value problem. Perturbation series approach to solving the Schrodinger equation. The eigenvalue problem

## Really Really Hard Probles

\begin{align} \label{eq:2ndODE} y'' + a(x)y' + b(x) y = 0 \end{align}

\begin{align} \label{eq:SchEQ} y'' + Q(x) y = 0 \end{align}

## Why So Hard?

first order linear equation $y' + a(x)y = b(x)$ 是一个 routine standard EASY problem, 因为它可以 用 integrating factor 来解. 下面说明, (\ref{eq:SchEQ})为什么 它 so hard

(Bender 在视频的 8:15 时说 I should take sort of 5 minutes and explain to you why it's diffcult, 然后在 26:43 时说完了 🤣 )

## Perturbation

\begin{align} \label{eq:SchEQPerturb} y'' + \epsilon Q(x) y = 0 \end{align}

(为什么加在这里? 因为 uperburbed problem 是可解的.) 考虑边界条件

\begin{align} y(0) =& \alpha \\ y'(0) =& \beta \end{align}

unpert: $y_0(x) = \alpha + \beta x$

Assuming $y(x) = \sum_{n=0}^{\infty} a_n(x)\epsilon^n$ 代加 (\ref{eq:SchEQPerturb})

\begin{align} \sum_{n=0}^{\infty} a_n''(x)\epsilon^n + \sum_{n=1}^{\infty} Q(x) a_{n-1}(x) \epsilon^n = 0 \end{align}

• $\epsilon^0$ : $a_0'' = 0$ , 这时无微扰时的解 $a_0(x) = \alpha + \beta x$
• $\epsilon^n$ : $a_n'' = -Q(x)a_{n-1}$ 因为 $a_0$ 已经满足边界条件了, 所以更高阶的边界条件全部为 $0$ , 也就是 $a_n(0) = a_n'(x) = 0$ , 那么积分就可以得到全部的系数
\begin{align} a_n'(x) =& -\int_0^x \mathrm{d}s\cdot Q(s)a_{n-1}(s) \\ a_n(x) =& -\int_0^x\mathrm{d}t \int_0^t \mathrm{d}s Q(s)a_{n-1}(s) \end{align}

\begin{align} a_n(x) = (-1)^n \int \int Q\int \int Q \cdots \int \int Q (\alpha + \beta x) \end{align}

Boys, we are powerful! We can do anything! :)

## Eigenvalue Problem

\begin{align} \left(-\frac{\mathrm{d}^2}{\mathrm{d}x^2} + \frac{x^2}{4} + \epsilon \frac{x^4}{4} \right)\psi = E( \epsilon )\psi \end{align}

\begin{align} E_{\mathrm{ground state}} = \frac{1}{2} + \frac{3}{4}\epsilon - \frac{21}{8}\epsilon^2 - \frac{333}{16}\epsilon^3 + \cdots \end{align}

You've been cheated! ... Everything you've been taught is garbage unless we can make sense out of this.

## What the Nature of the Singularity Is?

\begin{align} H = \left( \begin{array}{cc} a & 0 \\ 0 & b \end{array} \right) + \epsilon\left( \begin{array}{cc} 0 & c \\ c & 0 \end{array} \right) \end{align}
I look at this problem and I say 'oh, god! That's a hard problem to sovle! I think I'll use perturbation theory.'
\begin{align} E_{\mp}(\epsilon) = \frac{a + b \pm\sqrt{(a - b)^2 + 4 \epsilon^2 c^2}}{2} \end{align}

Remember this: quantum mechanics is not quantized. Because if I'am allowed to vary $\epsilon$ into the complexs plane, you smoothly go from one energy level to the other energy level.
Quantizations comes from counting the sheets in a Riemann surface.
Different energy levels are not independent numbers like you know.
If we enlarge our way of thinking about problem by doing perturbation theory and by introducing this magnificient parameter because, it gives a whole new way of thinking about the world!

## Summary

\begin{align} y'' + Q(x)y' + b(x) y = 0 \end{align}

\begin{align} y'' + Q(x) y = 0 \end{align}

Quantizations comes from counting the sheets in a Riemann surface.

Different energy levels are not independent numbers like you know.

## Reference

• 很好的讲解 integrating factor 的视频 First Order Linear Differential Equation & Integrating Factor (idea/strategy/example) by blackpenredpen , 有时间可以整理一下 note: https://www.youtube.com/watch?v=DJsjZ5aYK_g