Mathematical Physics by Carl Bender
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lecture 1 perturbation method
- solve <math>x^5+x=1</math>
method 1
- try <math>x^5+\epsilon x=1</math>
- find <math>x(\epsilon)</math> satisfying <math>x(\epsilon)^5+\epsilon x(\epsilon)=1</math>
- answer
- <math>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</math>
- Setting <math>\epsilon=1</math> gives numerical value <math>0.75\cdots</math>
weak coupling approach
- use the similar idea to Feynman diagrams
- try <math>\epsilon x^5+ x=1</math>
- we get
- <math>
x(\epsilon)=1-\epsilon +5 \epsilon ^2-35 \epsilon ^3+285 \epsilon ^4-2530 \epsilon ^5+23751 \epsilon ^6+\cdots </math>
- can we get a meaningful number out of this?
- yes, for example, Pade summation can be used
asymptotics
- <math>f\sim g\, \quad (x\to x_0)</math> iff :<math>\lim_{x\to x_0}\frac{f(x)}{g(x)}=1</math>
- apply the method of dominant balance to <math>\epsilon x^5+ x=1</math>
- <math>x^4\sim -1/\epsilon\, \quad (\epsilon \to 0)</math> and thus
- <math>x\sim \frac{\omega}{\epsilon^{1/4}}\, \quad (\epsilon \to 0)</math> where <math>\omega^4=-1</math>
- this is the first order approximation and we can have more terms
lecture 2
second order ordinary differential equation
- 틀:수학노트
- <math>y+Q(x)y=0</math> Schrodinger equation
- this is a very hard problem to solve
- consider a perturbed equation <math>y+\epsilon Q(x)y=0</math> so that its unperturbed equation is <math>y=0</math> with initial conditions <math>y(0)=\alpha, y'(0)=\beta</math>
- take the formal solution <math>y(x)=\sum_{n}a_n(x)\epsilon^n</math> where <math>a_0(x)=\alpha+\beta x</math>
- for it to be a solution, it should satisfy
- <math>
a_n(x)=-Q(x)a_{n-1}(x) </math> for each <math>n>0</math>
- thus we get
- <math>
a_n(x)=-\int_0^x\int_0^t Q(s)a_{n-1}(s)\,ds\,dt </math>
eigenvalue problem
- Schrodinger equation
- <math>
-\left(\frac{d^2}{dx^2} + V(x)\right)\psi=E \psi </math>
- if <math>V(x)=x^2/4</math>, we get harmonic oscillator
- anharmonic oscillator problem (similar to Phi-4 theory)
- <math>
-\left(\frac{d^2}{dx^2} +x^2/4+x^4/4\right)\psi=E \psi </math>
- perturbed version
- <math>
-\left(\frac{d^2}{dx^2} +x^2/4+\epsilon x^4/4\right)\psi(\epsilon)=E(\epsilon)\psi(\epsilon) </math>
- <math>E(\epsilon)=\sum_n a_n(x)\epsilon^n</math>
- <math>\psi(\epsilon)=\sum_n \psi_n(x)\epsilon^n</math>
- ground state <math>\psi_0(x)=e^{-x^2/2}</math> with <math>a_0=1/2</math>
Riemann surface and discrete spectrum
- analytic continuation using the parameter <math>\epsilon</math> 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.