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| 4번째 줄: | 4번째 줄: | ||
==lecture 1 perturbation method== | ==lecture 1 perturbation method== | ||
| − | * solve | + | * solve <math>x^5+x=1</math> |
===method 1=== | ===method 1=== | ||
| − | * try | + | * try <math>x^5+\epsilon x=1</math> |
| − | * find | + | * find <math>x(\epsilon)</math> satisfying <math>x(\epsilon)^5+\epsilon x(\epsilon)=1</math> |
* answer | * 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 | + | * Setting <math>\epsilon=1</math> gives numerical value <math>0.75\cdots</math> |
===weak coupling approach=== | ===weak coupling approach=== | ||
* use the similar idea to Feynman diagrams | * use the similar idea to Feynman diagrams | ||
| − | * try | + | * try <math>\epsilon x^5+ x=1</math> |
* we get | * we get | ||
| − | + | :<math> | |
x(\epsilon)=1-\epsilon +5 \epsilon ^2-35 \epsilon ^3+285 \epsilon ^4-2530 \epsilon ^5+23751 \epsilon ^6+\cdots | 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? | * can we get a meaningful number out of this? | ||
* yes, for example, Pade summation can be used | * yes, for example, Pade summation can be used | ||
| 25번째 줄: | 25번째 줄: | ||
===asymptotics=== | ===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 | + | * 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 | * this is the first order approximation and we can have more terms | ||
| 35번째 줄: | 35번째 줄: | ||
===second order ordinary differential equation=== | ===second order ordinary differential equation=== | ||
* {{수학노트|url=이계_선형_미분방정식}} | * {{수학노트|url=이계_선형_미분방정식}} | ||
| − | * | + | * <math>y''+Q(x)y=0</math> Schrodinger equation |
* this is a very hard problem to solve | * this is a very hard problem to solve | ||
| − | * consider a perturbed equation | + | * 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 | + | * 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 | * for it to be a solution, it should satisfy | ||
| − | + | :<math> | |
a_n''(x)=-Q(x)a_{n-1}(x) | a_n''(x)=-Q(x)a_{n-1}(x) | ||
| − | + | </math> | |
| − | for each | + | for each <math>n>0</math> |
* thus we get | * thus we get | ||
| − | + | :<math> | |
a_n(x)=-\int_0^x\int_0^t Q(s)a_{n-1}(s)\,ds\,dt | a_n(x)=-\int_0^x\int_0^t Q(s)a_{n-1}(s)\,ds\,dt | ||
| − | + | </math> | |
===eigenvalue problem=== | ===eigenvalue problem=== | ||
* Schrodinger equation | * Schrodinger equation | ||
| − | + | :<math> | |
-\left(\frac{d^2}{dx^2} + V(x)\right)\psi=E \psi | -\left(\frac{d^2}{dx^2} + V(x)\right)\psi=E \psi | ||
| − | + | </math> | |
| − | * if | + | * if <math>V(x)=x^2/4</math>, we get harmonic oscillator |
* anharmonic oscillator problem (similar to [[Phi-4 theory]]) | * 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 | -\left(\frac{d^2}{dx^2} +x^2/4+x^4/4\right)\psi=E \psi | ||
| − | + | </math> | |
* perturbed version | * perturbed version | ||
| − | + | :<math> | |
-\left(\frac{d^2}{dx^2} +x^2/4+\epsilon x^4/4\right)\psi(\epsilon)=E(\epsilon)\psi(\epsilon) | -\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 | + | * ground state <math>\psi_0(x)=e^{-x^2/2}</math> with <math>a_0=1/2</math> |
===Riemann surface and discrete spectrum=== | ===Riemann surface and discrete spectrum=== | ||
| − | * analytic continuation using the parameter | + | * analytic continuation using the parameter <math>\epsilon</math> gives all the energy states |
* they correspond to different sheets of a Riemann surface | * they correspond to different sheets of a Riemann surface | ||
| 84번째 줄: | 84번째 줄: | ||
[[분류:talks and lecture notes]] | [[분류:talks and lecture notes]] | ||
| + | [[분류:migrate]] | ||
2020년 11월 13일 (금) 09:05 기준 최신판
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.