"Quantum scattering"의 두 판 사이의 차이
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imported>Pythagoras0 잔글 (찾아 바꾸기 – “<h5>” 문자열을 “==” 문자열로) |
imported>Pythagoras0 잔글 (찾아 바꾸기 – “</h5>” 문자열을 “==” 문자열로) |
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| 1번째 줄: | 1번째 줄: | ||
| − | ==introduction | + | ==introduction== |
* <math>\varphi_{xx}+(\lambda-u)\varphi=0</math> | * <math>\varphi_{xx}+(\lambda-u)\varphi=0</math> | ||
| 10번째 줄: | 10번째 줄: | ||
| − | ==time independent Schrodinger equation | + | ==time independent Schrodinger equation== |
* [[Schrodinger equation]]<br><math>E \psi = -\frac{\hbar^2}{2m}{\partial^2 \psi \over \partial x^2} + V(x)\psi</math><br> | * [[Schrodinger equation]]<br><math>E \psi = -\frac{\hbar^2}{2m}{\partial^2 \psi \over \partial x^2} + V(x)\psi</math><br> | ||
| 17번째 줄: | 17번째 줄: | ||
| − | ==continuous spectrum | + | ==continuous spectrum== |
* e^{ikx} represents a wave traveling to the right, and e^{−ikx} one traveling to the left | * e^{ikx} represents a wave traveling to the right, and e^{−ikx} one traveling to the left | ||
| 25번째 줄: | 25번째 줄: | ||
| − | ==potential scattering | + | ==potential scattering== |
<math>r=t-1</math> | <math>r=t-1</math> | ||
| 37번째 줄: | 37번째 줄: | ||
| − | ==delta potential example | + | ==delta potential example== |
* [[delta potential scattering]] | * [[delta potential scattering]] | ||
| 45번째 줄: | 45번째 줄: | ||
| − | ==harmonic oscillator | + | ==harmonic oscillator== |
* [[harmonic oscillator in quantum mechanics]] | * [[harmonic oscillator in quantum mechanics]] | ||
| 53번째 줄: | 53번째 줄: | ||
| − | ==sech potential example | + | ==sech potential example== |
* [[sech potential example]] | * [[sech potential example]] | ||
| 67번째 줄: | 67번째 줄: | ||
| − | ==related items | + | ==related items== |
* [[inverse scattering method]] | * [[inverse scattering method]] | ||
| 75번째 줄: | 75번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">encyclopedia | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">encyclopedia== |
* [http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation http://en.wikipedia.org/wiki/Schrödinger_equation]<br> | * [http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation http://en.wikipedia.org/wiki/Schrödinger_equation]<br> | ||
2012년 10월 28일 (일) 15:37 판
introduction
- \(\varphi_{xx}+(\lambda-u)\varphi=0\)
- discrete spectrum \(\lambda<0\)
- continuous spectrum \(\lambda>0\)
- for lists http://en.wikipedia.org/wiki/Delta_potential_barrier_(QM)
time independent Schrodinger equation
- Schrodinger equation
\(E \psi = -\frac{\hbar^2}{2m}{\partial^2 \psi \over \partial x^2} + V(x)\psi\) - simplified form
\(-\varphi_{xx}+u(x)\varphi = \lambda\varphi\)
\(\varphi_{xx}+(\lambda-u(x))\varphi=0\)
continuous spectrum
- e^{ikx} represents a wave traveling to the right, and e^{−ikx} one traveling to the left
- e^{−ikx} is incoming wave from the right to the left
- reflection and transmission coefficient
\(\varphi \sim e^{-ikx}+\rho(k,t)e^{ikx}\) as \(x\to +\infty\)
\(\varphi \sim \tau(k,t)e^{-ikx}\) as \(x\to -\infty\)
\(\rho(k,t)\) and \(\tau(k,t)\) are called the reflection and transmission coefficient
potential scattering
\(r=t-1\)
If t is of the form \(t=\frac{1}{1-ai}\) (real number a), then
\(|r|^2+|t|^2=1\)
delta potential example
harmonic oscillator
sech potential example
- \[Lambda] := -1
u[x_] := -2 Sech[x]^2
f[x_] := Sech[x]
Simplify[D[D[f[x], x], x] + (\[Lambda] - u[x]) f[x]]
Plot[u[x], {x, -5, 5}]