"Quantum scattering"의 두 판 사이의 차이
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* continuous spectrum <math>\lambda>0</math> | * continuous spectrum <math>\lambda>0</math> | ||
* for lists [http://en.wikipedia.org/wiki/Delta_potential_barrier_%28QM%29 http://en.wikipedia.org/wiki/Delta_potential_barrier_(QM)] | * for lists [http://en.wikipedia.org/wiki/Delta_potential_barrier_%28QM%29 http://en.wikipedia.org/wiki/Delta_potential_barrier_(QM)] | ||
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* [[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> | ||
* simplified form<br><math>-\varphi_{xx}+u(x)\varphi = \lambda\varphi</math><br><math>\varphi_{xx}+(\lambda-u(x))\varphi=0</math><br> | * simplified form<br><math>-\varphi_{xx}+u(x)\varphi = \lambda\varphi</math><br><math>\varphi_{xx}+(\lambda-u(x))\varphi=0</math><br> | ||
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| − | <h5>example</h5> | + | <h5>sech potential example</h5> |
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# \[Lambda] := -1<br> u[x_] := -2 Sech[x]^2<br> f[x_] := Sech[x]<br> Simplify[D[D[f[x], x], x] + (\[Lambda] - u[x]) f[x]]<br> Plot[u[x], {x, -5, 5}] | # \[Lambda] := -1<br> u[x_] := -2 Sech[x]^2<br> f[x_] := Sech[x]<br> Simplify[D[D[f[x], x], x] + (\[Lambda] - u[x]) f[x]]<br> Plot[u[x], {x, -5, 5}] | ||
2011년 2월 14일 (월) 08:36 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}]
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