Mathematical Physics by Carl Bender
imported>Pythagoras0님의 2020년 11월 13일 (금) 08:05 판
list
lecture 1 perturbation method
- solve $x^5+x=1$
method 1
- try $x^5+\epsilon x=1$
- find $x(\epsilon)$ satisfying $x(\epsilon)^5+\epsilon x(\epsilon)=1$
- answer
$$x(\epsilon)=1-\frac{\epsilon }{5}-\frac{\epsilon ^2}{25}-\frac{\epsilon ^3}{125}+\frac{21 \epsilon ^5}{15625}+\frac{78 \epsilon ^6}{78125}+\cdots$$
- Setting $\epsilon=1$ gives numerical value $0.75\cdots$
weak coupling approach
- use the similar idea to Feynman diagrams
- try $\epsilon x^5+ x=1$
- we get
$$ x(\epsilon)=1-\epsilon +5 \epsilon ^2-35 \epsilon ^3+285 \epsilon ^4-2530 \epsilon ^5+23751 \epsilon ^6+\cdots $$
- can we get a meaningful number out of this?
- yes, for example, Pade summation can be used
asymptotics
- $f\sim g\, \quad (x\to x_0)$ iff $$\lim_{x\to x_0}\frac{f(x)}{g(x)}=1$$
- apply the method of dominant balance to $\epsilon x^5+ x=1$
- $x^4\sim -1/\epsilon\, \quad (\epsilon \to 0)$ and thus
$$x\sim \frac{\omega}{\epsilon^{1/4}}\, \quad (\epsilon \to 0)$$ where $\omega^4=-1$
- this is the first order approximation and we can have more terms
lecture 2
second order ordinary differential equation
- 틀:수학노트
- $y+Q(x)y=0$ Schrodinger equation
- this is a very hard problem to solve
- consider a perturbed equation $y+\epsilon Q(x)y=0$ so that its unperturbed equation is $y=0$ with initial conditions $y(0)=\alpha, y'(0)=\beta$
- take the formal solution $y(x)=\sum_{n}a_n(x)\epsilon^n$ where $a_0(x)=\alpha+\beta x$
- for it to be a solution, it should satisfy
$$ a_n''(x)=-Q(x)a_{n-1}(x) $$ for each $n>0$
- thus we get
$$ a_n(x)=-\int_0^x\int_0^t Q(s)a_{n-1}(s)\,ds\,dt $$
eigenvalue problem
- Schrodinger equation
$$ -\left(\frac{d^2}{dx^2} + V(x)\right)\psi=E \psi $$
- if $V(x)=x^2/4$, we get harmonic oscillator
- anharmonic oscillator problem (similar to Phi-4 theory)
$$ -\left(\frac{d^2}{dx^2} +x^2/4+x^4/4\right)\psi=E \psi $$
- perturbed version
$$ -\left(\frac{d^2}{dx^2} +x^2/4+\epsilon x^4/4\right)\psi(\epsilon)=E(\epsilon)\psi(\epsilon) $$
- $E(\epsilon)=\sum_n a_n(x)\epsilon^n$
- $\psi(\epsilon)=\sum_n \psi_n(x)\epsilon^n$
- ground state $\psi_0(x)=e^{-x^2/2}$ with $a_0=1/2$
Riemann surface and discrete spectrum
- analytic continuation using the parameter $\epsilon$ gives all the energy states
- they correspond to different sheets of a Riemann surface
lecture 3
- Shanks transform for alternating series
- two examples
computational resource
books
- Bender, Carl M., and Steven A. Orszag. 1999. Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory. Springer.