"Quantum scattering"의 두 판 사이의 차이

수학노트
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<h5>introduction</h5>
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==introduction==
  
 
* <math>\varphi_{xx}+(\lambda-u)\varphi=0</math>
 
* <math>\varphi_{xx}+(\lambda-u)\varphi=0</math>
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* discrete spectrum <math>\lambda<0</math>
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* discrete spectrum <math>\lambda<0</math>
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* continuous spectrum <math>\lambda>0</math>
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* 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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==time independent Schrodinger equation==
  
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* [[Schrodinger equation]]
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:<math>E \psi = -\frac{\hbar^2}{2m}{\partial^2 \psi \over \partial x^2} + V(x)\psi</math>
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* simplified form
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:<math>-\varphi_{xx}+u(x)\varphi = \lambda\varphi</math>
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:<math>\varphi_{xx}+(\lambda-u(x))\varphi=0</math>
  
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<h5>continuous spectrum</h5>
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* e^{ikx} represents a wave traveling to the right, and e^{−ikx} one traveling to the left
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==continuous spectrum==
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* e^{−ikx} is incoming wave from the right to the left
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* <math>e^{−ikx}</math> is incoming wave from the right to the left
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* 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>e^{ikx}</math> represents a wave traveling to the right
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* reflection and transmission coefficient  
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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
  
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==potential scattering==
  
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<h5>time independent Schrodinger equation</h5>
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<math>r=t-1</math>
  
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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>
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If t is of the form <math>t=\frac{1}{1-ai}</math> (real number a), then
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*  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>
 
  
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<math>|r|^2+|t|^2=1</math>
  
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<h5>delta potential example</h5>
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* 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>
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*  we impose two conditions on the wave function<br>
 
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**  the wave function be continuous in the origin
 
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**  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
 
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*  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>
 
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*  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>
 
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*  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>
 
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* special case of scattering problem
 
  
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==delta potential example==
  
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* [[delta potential scattering]]
  
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<h5>harmonic oscillator</h5>
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==harmonic oscillator==
  
 
* [[harmonic oscillator in quantum mechanics]]
 
* [[harmonic oscillator in quantum mechanics]]
  
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<h5>example</h5>
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==sech potential example==
  
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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}]
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* [[sech potential example]]
  
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==related items==
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<h5>history</h5>
 
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
 
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<h5>related items</h5>
 
  
 
* [[inverse scattering method]]
 
* [[inverse scattering method]]
  
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==expositions==
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* http://docs.google.com/viewer?a=v&q=cache:nCR9E6bwofAJ:www.rpi.edu/dept/phys/courses/phys410/lct11.pdf+plane+wave+scattering+potential&hl=ko&gl=us&pid=bl&srcid=ADGEEShoNleR3WGnxKKLrDSg_ZNAlytq0EsPPn2ZI2GN79gnfdrNls8jrHdLk68yNQnq4RhMdJdTJ25r52naDFkQcYK9jLXMI7awu5BGD2GvPj05Ky5ZQTu0cKdZVyvI_Ff4rcbrIy7D&sig=AHIEtbSbvDC8BEFTsMXaQRZn04q-bivLnQ
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* http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/ScatteringTheory.htm
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<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>
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==encyclopedia==
  
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* http://en.wikipedia.org/wiki/
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* [http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation http://en.wikipedia.org/wiki/Schrödinger_equation]
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* [http://en.wikipedia.org/wiki/Delta_potential_barrier_%28QM%29 http://en.wikipedia.org/wiki/Delta_potential_barrier_(QM)]
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* [http://en.wikipedia.org/wiki/Spectrum_%28functional_analysis%29 http://en.wikipedia.org/wiki/Spectrum_(functional_analysis)]
 
* http://en.wikipedia.org/wiki/Rectangular_potential_barrier
 
* http://en.wikipedia.org/wiki/Rectangular_potential_barrier
 
* http://en.wikipedia.org/wiki/Step_potential
 
* http://en.wikipedia.org/wiki/Step_potential
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* http://www.scholarpedia.org/
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[[분류:physics]]
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* [http://eom.springer.de/ http://eom.springer.de]
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[[분류:math and physics]]
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* http://www.proofwiki.org/wiki/
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[[분류:migrate]]
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* Princeton companion to mathematics([[2910610/attachments/2250873|Companion_to_Mathematics.pdf]])
 
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<h5>books</h5>
 
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* [[2011년 books and articles]]
 
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* http://library.nu/search?q=
 
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* http://library.nu/search?q=
 
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<h5>expositions</h5>
 
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<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%;">articles</h5>
 
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* [http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation http://en.wikipedia.org/wiki/Schrödinger_equation]<br>
 
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* [http://en.wikipedia.org/wiki/Spectrum_%28functional_analysis%29 http://en.wikipedia.org/wiki/Spectrum_(functional_analysis)]<br>
 
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* http://www.ams.org/mathscinet
 
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* http://www.zentralblatt-math.org/zmath/en/
 
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* http://arxiv.org/
 
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* http://www.pdf-search.org/
 
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* http://pythagoras0.springnote.com/
 
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* [http://math.berkeley.edu/%7Ereb/papers/index.html http://math.berkeley.edu/~reb/papers/index.html]
 
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* http://dx.doi.org/
 
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<h5>question and answers(Math Overflow)</h5>
 
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* http://mathoverflow.net/search?q=
 
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* http://mathoverflow.net/search?q=
 
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<h5>blogs</h5>
 
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*  구글 블로그 검색<br>
 
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**  http://blogsearch.google.com/blogsearch?q=<br>
 
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** http://blogsearch.google.com/blogsearch?q=
 
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* http://ncatlab.org/nlab/show/HomePage
 
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<h5>experts on the field</h5>
 
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* http://arxiv.org/
 
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<h5>links</h5>
 
  
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* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
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==메타데이터==
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* [http://pythagoras0.springnote.com/pages/1947378 수식표현 안내]
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===위키데이터===
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* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
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* ID :  [https://www.wikidata.org/wiki/Q2381860 Q2381860]
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* http://functions.wolfram.com/
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===Spacy 패턴 목록===
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* [{'LOWER': 'delta'}, {'LEMMA': 'potential'}]

2021년 2월 17일 (수) 03:03 기준 최신판

introduction


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

  • \(e^{−ikx}\) is incoming wave from the right to the left
  • \(e^{ikx}\) represents a wave traveling to the right
  • 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


related items


expositions


encyclopedia

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'delta'}, {'LEMMA': 'potential'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=Quantum_scattering&oldid=51892"