"Quantum scattering"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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12번째 줄: | 12번째 줄: | ||
==time independent Schrodinger equation== | ==time independent Schrodinger equation== | ||
− | * [[Schrodinger equation]] | + | * [[Schrodinger equation]] |
− | * | + | :<math>E \psi = -\frac{\hbar^2}{2m}{\partial^2 \psi \over \partial x^2} + V(x)\psi</math> |
+ | * simplified form | ||
+ | :<math>-\varphi_{xx}+u(x)\varphi = \lambda\varphi</math> | ||
+ | :<math>\varphi_{xx}+(\lambda-u(x))\varphi=0</math> | ||
2013년 2월 19일 (화) 14:39 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
\[E \psi = -\frac{\hbar^2}{2m}{\partial^2 \psi \over \partial x^2} + V(x)\psi\]
- 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