"Harmonic oscillator in quantum mechanics"의 두 판 사이의 차이
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* 해밀토니안<br><math>H(P,X)=\frac{P^2}{2m}+\frac{m}{2}\omega^{2}X^2</math><br> | * 해밀토니안<br><math>H(P,X)=\frac{P^2}{2m}+\frac{m}{2}\omega^{2}X^2</math><br> | ||
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<h5>Schrodinger equation</h5> | <h5>Schrodinger equation</h5> | ||
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2011년 2월 8일 (화) 03:30 판
introduction
harmonic oscillator in classical mechanics
- 고전역학에서의 조화진동자(고전역학에서의 가적분성 항목 참조)
- 질량 m, frequency \(\omega\) 인 조화진동자
- 해밀토니안
\(H(p,q)=\frac{p^2}{2m}+\frac{m}{2}\omega^{2}q^2\) - 해밀턴 방정식
\(\dot{q}=\partial H/\partial p=\frac{p}{m}\)
\(\dot{p}=-\partial H/\partial q=-m\omega^{2}q\) - 운동방정식
\(\ddot{x}=-\omega^{2} x\) 즉 \(\ddot{x}+\omega^{2} x=0\)
quantum harmonic oscillator
- 해밀토니안
\(H(P,X)=\frac{P^2}{2m}+\frac{m}{2}\omega^{2}X^2\)
creation and annhilation operators
- the position operators and momentum operators satisfy the relation
\([X,P] = X P - P X = i \hbar\)Heisenberg group and Heisenberg algebra - define operators as follows
\(a =\sqrt{m\omega \over 2\hbar} \left(x + {i \over m \omega} p \right)\)
\(a^{\dagger} =\sqrt{m \omega \over 2\hbar} \left( x - {i \over m \omega} p \right)\) - Hamiltonian
\(H = \hbar \omega \left(a^{\dagger}a + 1/2\right)\) - Commutation relation
\(\left[a , a^{\dagger} \right] = 1\)
\(\left[ H, a \right]= - \hbar \omega a\)
\(\left[ H, a^\dagger \right] = \hbar \omega a^\dagger\)
energy eigenstates
- Assume that Planck’s constant equals 1
- a harmonic oscillator that vibrates with frequency \(\omega\) can have energy \(\frac{\omega}{2}, (1 +\frac{1}{2})\omega, (2 +\frac{1}{2})\omega,(3 +\frac{1}{2})\omega,\cdots\) in units where
- The lowest energy is not zero! It’s \(\omega/2\). This is called the ground state energy of the oscillator.
Schrodinger equation
path integral formulation
Groundstate correlation functions
Green's funtion
history
encyclopedia
- http://ko.wikipedia.org/wiki/양자조화진동자
- http://en.wikipedia.org/wiki/Quantum_harmonic_oscillator
- http://en.wikipedia.org/wiki/
- http://www.scholarpedia.org/
- http://www.proofwiki.org/wiki/
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
- 2010년 books and articles
- http://gigapedia.info/1/
- http://gigapedia.info/1/
- http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
expositions
- http://web.physik.rwth-aachen.de/~meden/vielteilchenneu/skriptka2.pdf
- http://www.odu.edu/~jdudek/lecture_notes/GradQM_Second_Semester/ProblemSet0.pdf
articles
- 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
- 구글 블로그 검색
- http://ncatlab.org/nlab/show/HomePage
experts on the field