"Quantum scattering"의 두 판 사이의 차이
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| 14번째 줄: | 14번째 줄: | ||
<h5>continuous spectrum</h5> | <h5>continuous spectrum</h5> | ||
| − | * 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 |
| − | * | + | * e^{−ikx} is incoming wave from the right to the left |
| − | * | + | * reflection and transmission coefficient<br><math>\varphi \sim e^{-ikx}+\rho(k,t)e^{ikx}</math> as <math>x\to +\infty</math><br><math>\varphi \sim \tau(k,t)e^{-ikx}</math> as <math>x\to -\infty</math><br><math>\rho(k,t)</math> and <math>\tau(k,t)</math> are called the reflection and transmission coefficient<br> |
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| − | <math>\varphi \sim e^{-ikx}+\rho(k,t)e^{ikx}</math> as <math>x\to +\infty</math> | ||
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| − | <math>\varphi \sim \tau(k,t)e^{-ikx}</math> as <math>x\to -\infty</math> | ||
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| − | <math>\rho(k,t)</math> and <math>\tau(k,t)</math> are called the reflection and transmission coefficient | ||
| 32번째 줄: | 25번째 줄: | ||
* [[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> | ||
| − | * <math>\varphi_{xx}+(\lambda-u)\varphi=0</math> | + | * simplified form<br><math>-\varphi_{xx}+u(x)\varphi = \lambda\varphi</math><br> <br><math>\varphi_{xx}+(\lambda-u(x))\varphi=0</math><br> <br> |
| − | + | <h5>delta potential example</h5> | |
| − | < | + | <math>V(x) = \lambda\delta(x)</math> |
| − | + | <math>\psi(x) = \begin{cases} \psi_{\mathrm L}(x) = A_{\mathrm r}e^{ikx} + A_{\mathrm l}e^{-ikx}, & \text{ if } x<0; \\ \psi_{\mathrm R}(x) = B_{\mathrm r}e^{ikx} + B_{\mathrm l}e^{-ikx}, & \text{ if } x>0, \end{cases}</math> | |
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2011년 2월 8일 (화) 12:37 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)
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
time independent Schrodinger equation
- Schrodinger equation
\(E \psi = -\frac{\hbar^2}{2m}{\partial^2 \psi \over \partial x^2} + V(x)\psi\) - simplified form
\(-\varphi_{xx}+u(x)\varphi = \lambda\varphi\)
\(\varphi_{xx}+(\lambda-u(x))\varphi=0\)
delta potential example
\(V(x) = \lambda\delta(x)\)
\(\psi(x) = \begin{cases} \psi_{\mathrm L}(x) = A_{\mathrm r}e^{ikx} + A_{\mathrm l}e^{-ikx}, & \text{ if } x<0; \\ \psi_{\mathrm R}(x) = B_{\mathrm r}e^{ikx} + B_{\mathrm l}e^{-ikx}, & \text{ if } x>0, \end{cases}\)
harmonic oscillator
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}]
history
encyclopedia
- http://en.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/Delta_potential_barrier_(QM)
- http://en.wikipedia.org/wiki/Rectangular_potential_barrier
- http://en.wikipedia.org/wiki/Step_potential
- http://www.scholarpedia.org/
- http://eom.springer.de
- http://www.proofwiki.org/wiki/
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
expositions
articles
- http://en.wikipedia.org/wiki/Schrödinger_equation
- http://en.wikipedia.org/wiki/Spectrum_(functional_analysis)
- http://www.ams.org/mathscinet
- http://www.zentralblatt-math.org/zmath/en/
- http://arxiv.org/
- http://www.pdf-search.org/
- http://pythagoras0.springnote.com/
- http://math.berkeley.edu/~reb/papers/index.html
- http://dx.doi.org/
question and answers(Math Overflow)
blogs
- 구글 블로그 검색
- http://ncatlab.org/nlab/show/HomePage
experts on the field