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

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<h5>creation and annhilation operators</h5>
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==path integral formulation==
  
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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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* Groundstate correlation functions
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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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* [[path integral formulation of quantum mechanics]]
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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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==Green's funtion==
  
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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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==related items==
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<h5>Schrodinger equation</h5>
 
  
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* [[Heisenberg group and Heisenberg algebra]]
 
* [[Schrodinger equation]]
 
* [[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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<math>V(x)=\frac{k}{2}x^2</math>
 
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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>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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==expositions==
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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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* [{'LOWER': 'quantum'}, {'LOWER': 'harmonic'}, {'LEMMA': 'oscillator'}]

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'}]
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