"Quantum scattering"의 두 판 사이의 차이

2011년 2월 14일 (월) 09:34 판

Schrodinger equation
\(E \psi = -\frac{\hbar^2}{2m}{\partial^2 \psi \over \partial x^2} + V(x)\psi\)
simplified form
\(-\varphi_{xx}+u(x)\varphi = \lambda\varphi\)
\(\varphi_{xx}+(\lambda-u(x))\varphi=0\)

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

\(r=t-1\)

If t is of the form \(t=\frac{1}{1-ai}\) (real number a), then

\(|r|^2+|t|^2=1\)

\[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}]

@@ 55번째 줄: / 55번째 줄: @@
 <h5>sech potential example</h5>
+* [[sech potential example]]
-#
 # \[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}]
@@ 86번째 줄: / 85번째 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>
-* http://en.wikipedia.org/wiki/
+*   <br>
-* [http://en.wikipedia.org/wiki/Delta_potential_barrier_%28QM%29 http://en.wikipedia.org/wiki/Delta_potential_barrier_(QM)]
+* http://en.wikipedia.org/wiki/[http://en.wikipedia.org/wiki/Delta_potential_barrier_%28QM%29 ]
 * http://en.wikipedia.org/wiki/Rectangular_potential_barrier
 * http://en.wikipedia.org/wiki/Step_potential