"Harmonic oscillator in quantum mechanics"의 두 판 사이의 차이

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<h5>introduction</h5>
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==path integral formulation==
  
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* Groundstate correlation functions
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* [[path integral formulation of quantum mechanics]]
  
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<h5>harmonic oscillator in classical mechanics</h5>
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==Green's funtion==
  
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* 고전역학에서의 조화진동자([http://statphys.springnote.com/pages/5695329 고전역학에서의 가적분성] 항목 참조)
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* 질량 m, frequency <math>\omega</math> 인 조화진동자<br>
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*  해밀토니안<br><math>H(p,q)=\frac{p^2}{2m}+\frac{m}{2}\omega^{2}q^2</math><br>
 
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*  해밀턴 방정식<br><math>\dot{q}=\partial H/\partial p=\frac{p}{m}</math><br><math>\dot{p}=-\partial H/\partial q=-m\omega^{2}q</math><br>
 
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*  운동방정식<br><math>\ddot{x}=-\omega^{2} x</math> 즉 <math>\ddot{x}+\omega^{2} x=0</math><br>
 
  
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==related items==
  
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* [[Heisenberg group and Heisenberg algebra]]
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* [[Schrodinger equation]]
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<h5>quantum harmonic oscillator</h5>
 
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<math>H(P,X)=\frac{P^2}{2m}+\frac{m}{2}\omega^{2}X^2</math>
 
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<h5>creation and annhilation operators</h5>
 
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* the position operators and momentum operators satisfy the relation<br><math>[X,P] = X P - P X = i \hbar</math>[[Heisenberg group and Heisenberg algebra]]<br>
 
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*  define operators as follows<br><math>a =\sqrt{m\omega \over 2\hbar} \left(x + {i \over m \omega} p \right)</math><br><math>a^{\dagger} =\sqrt{m \omega \over 2\hbar} \left( x - {i \over m \omega} p \right)</math><br>
 
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*  Hamiltonian<br><math>H = \hbar \omega \left(a^{\dagger}a + 1/2\right)</math><br>
 
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* Commutation relation<br><math>\left[a , a^{\dagger} \right] = 1</math><br><math>\left[ H, a \right]= - \hbar \omega a</math><br><math>\left[ H, a^\dagger \right] =  \hbar \omega a^\dagger</math><br>
 
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<h5>energy  eigenstates</h5>
 
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*  Assume that Planck’s constant equals 1<br>
 
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*  a harmonic oscillator that vibrates with frequency <math>\omega</math> can have energy <math>\frac{\omega}{2}, (1 +\frac{1}{2})\omega, (2 +\frac{1}{2})\omega,(3 +\frac{1}{2})\omega,\cdots</math> in units where<br>
 
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*  The lowest energy is not zero! It’s <math>\omega/2</math>. This is called the ground state energy of the oscillator.<br>
 
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<h5>Schrodinger equation</h5>
 
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*  
 
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<h5>path integral formulation</h5>
 
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Groundstate correlation functions
 
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<h5>Green's funtion</h5>
 
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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>
 
  
 
* [[free massless boson]]
 
* [[free massless boson]]
 
* [[partition function in string theory]]
 
* [[partition function in string theory]]
  
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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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* [http://ko.wikipedia.org/wiki/%EC%96%91%EC%9E%90%EC%A1%B0%ED%99%94%EC%A7%84%EB%8F%99%EC%9E%90 http://ko.wikipedia.org/wiki/양자조화진동자]
 
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* http://en.wikipedia.org/wiki/Quantum_harmonic_oscillator
 
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* http://en.wikipedia.org/wiki/
 
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* http://www.scholarpedia.org/
 
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* http://www.proofwiki.org/wiki/
 
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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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* [[2010년 books and articles]]<br>
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==expositions==
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* http://gigapedia.info/1/
 
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* http://gigapedia.info/1/
 
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* http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
 
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<h5>expositions</h5>
 
  
 
* [http://web.physik.rwth-aachen.de/%7Emeden/vielteilchenneu/skriptka2.pdf http://web.physik.rwth-aachen.de/~meden/vielteilchenneu/skriptka2.pdf]
 
* [http://web.physik.rwth-aachen.de/%7Emeden/vielteilchenneu/skriptka2.pdf http://web.physik.rwth-aachen.de/~meden/vielteilchenneu/skriptka2.pdf]
 
* [http://www.odu.edu/%7Ejdudek/lecture_notes/GradQM_Second_Semester/ProblemSet0.pdf http://www.odu.edu/~jdudek/lecture_notes/GradQM_Second_Semester/ProblemSet0.pdf]
 
* [http://www.odu.edu/%7Ejdudek/lecture_notes/GradQM_Second_Semester/ProblemSet0.pdf http://www.odu.edu/~jdudek/lecture_notes/GradQM_Second_Semester/ProblemSet0.pdf]
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[[분류:physics]]
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[[분류:math and physics]]
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[[분류:migrate]]
  
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==메타데이터==
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===위키데이터===
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* ID : [https://www.wikidata.org/wiki/Q677864 Q677864]
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===Spacy 패턴 목록===
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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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* [{'LOWER': 'quantum'}, {'LOWER': 'harmonic'}, {'LEMMA': 'oscillator'}]
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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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* [http://pythagoras0.springnote.com/pages/1947378 수식표현 안내]
 
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* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
 
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* http://functions.wolfram.com/
 

2021년 2월 17일 (수) 02:40 기준 최신판

path integral formulation



Green's funtion

related items




expositions

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'quantum'}, {'LOWER': 'harmonic'}, {'LEMMA': 'oscillator'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=Harmonic_oscillator_in_quantum_mechanics&oldid=51815"