"Quantum scattering"의 두 판 사이의 차이
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33번째 줄: | 33번째 줄: | ||
* Let the potential is given by <math>V(x) = \lambda\delta(x)</math><br><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><br> | * Let the potential is given by <math>V(x) = \lambda\delta(x)</math><br><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><br> | ||
* we impose two conditions on the wave function<br> | * we impose two conditions on the wave function<br> | ||
− | ** the wave function be continuous in the origin | + | ** the wave function be continuous in the origin |
** integrate the Schrödinger equation around x = 0, over an interval [−ε, +ε] and In the limit as ε → 0, the right-hand side of this equation vanishes; the left-hand side becomes | ** integrate the Schrödinger equation around x = 0, over an interval [−ε, +ε] and In the limit as ε → 0, the right-hand side of this equation vanishes; the left-hand side becomes | ||
− | * first condition<br> | + | * first condition<br><math>\psi(0) =\psi_L(0) = \psi_R(0) = A_r + A_l = B_r + B_l</math><br><math>A_r + A_l - B_r - B_l = 0</math><br> |
− | * second condition<br><math> -\frac{\hbar^2}{2 m} \int_{-\epsilon}^{+\epsilon} \psi''(x) \,dx + \int_{-\epsilon}^{+\epsilon} V(x)\psi(x) \,dx = E \int_{-\epsilon}^{+\epsilon} \psi(x) \,dx</math><br> LHS becomes <math>-\frac{\hbar^2}{2m}[\psi_R'(0)-\psi_L'(0)] +\lambda\psi(0)</math><br> RHS becomes<br> | + | * second condition<br><math> -\frac{\hbar^2}{2 m} \int_{-\epsilon}^{+\epsilon} \psi''(x) \,dx + \int_{-\epsilon}^{+\epsilon} V(x)\psi(x) \,dx = E \int_{-\epsilon}^{+\epsilon} \psi(x) \,dx</math><br> LHS becomes <math>-\frac{\hbar^2}{2m}[\psi_R'(0)-\psi_L'(0)] +\lambda\psi(0)</math><br> RHS becomes 0<br><math>-A_r + A_l + B_r - B_l =\frac{2m\lambda}{ik\hbar^2}(A_r + A_l)</math><br> |
+ | * the coefficient must satisfy<br><math>A_r + A_l - B_r - B_l = 0</math><br><math>-A_r + A_l + B_r - B_l =\frac{2m\lambda}{ik\hbar^2}(A_r + A_l)</math><br> | ||
+ | * special case of scattering problem | ||
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2011년 2월 8일 (화) 12:49 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
- Let the potential is given by \(V(x) = \lambda\delta(x)\)
\(\psi(x) = \begin{cases} \psi_{\mathrm L}(x) = A_{\mathrm r}e^{ikx} + A_{\mathrm l}e^{-ikx}, & \text{ if } x<0; \\ \psi_{\mathrm R}(x) = B_{\mathrm r}e^{ikx} + B_{\mathrm l}e^{-ikx}, & \text{ if } x>0, \end{cases}\) - we impose two conditions on the wave function
- the wave function be continuous in the origin
- integrate the Schrödinger equation around x = 0, over an interval [−ε, +ε] and In the limit as ε → 0, the right-hand side of this equation vanishes; the left-hand side becomes
- first condition
\(\psi(0) =\psi_L(0) = \psi_R(0) = A_r + A_l = B_r + B_l\)
\(A_r + A_l - B_r - B_l = 0\) - second condition
\( -\frac{\hbar^2}{2 m} \int_{-\epsilon}^{+\epsilon} \psi''(x) \,dx + \int_{-\epsilon}^{+\epsilon} V(x)\psi(x) \,dx = E \int_{-\epsilon}^{+\epsilon} \psi(x) \,dx\)
LHS becomes \(-\frac{\hbar^2}{2m}[\psi_R'(0)-\psi_L'(0)] +\lambda\psi(0)\)
RHS becomes 0
\(-A_r + A_l + B_r - B_l =\frac{2m\lambda}{ik\hbar^2}(A_r + A_l)\) - the coefficient must satisfy
\(A_r + A_l - B_r - B_l = 0\)
\(-A_r + A_l + B_r - B_l =\frac{2m\lambda}{ik\hbar^2}(A_r + A_l)\) - special case of scattering problem
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
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- http://ncatlab.org/nlab/show/HomePage
experts on the field