"Harmonic oscillator in quantum mechanics"의 두 판 사이의 차이
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| 22번째 줄: | 22번째 줄: | ||
<h5>creation and annhilation operators</h5> | <h5>creation and annhilation operators</h5> | ||
| − | <math>a =\sqrt{m\omega \over 2\hbar} \left(x + {i \over m \omega} p \right)</math> | + | * 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> |
| − | + | * 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> | |
| − | <math>a^{\dagger} =\sqrt{m \omega \over 2\hbar} \left( x - {i \over m \omega} p \right)</math> | + | * Hamiltonian<br><math>H = \hbar \omega \left(a^{\dagger}a + 1/2\right)</math><br> |
| − | + | * 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> | |
| − | <math>H = \hbar \omega \left(a^{\dagger}a + 1/2\right)</math> | ||
| − | |||
| − | <math>\left[a , a^{\dagger} \right] = 1</math> | ||
| 34번째 줄: | 31번째 줄: | ||
| − | <h5> | + | <h5>energy eigenstates</h5> |
| − | * | + | * Assume that Planck’s constant equals 1<br> |
| − | The lowest energy is not zero! It’s <math>\omega/2</math>. This is called the ground state energy of the oscillator. | + | * a harmonic oscillator that vibrates with frequency <math>\omega</math> can have energy <math>1/2\omega, (1 +1/2)\omega, (2 +1/2)\omega,(3 +1/2)\omega,\cdots</math> in units where<br> |
| + | * The lowest energy is not zero! It’s <math>\omega/2</math>. This is called the ground state energy of the oscillator.<br> | ||
2010년 9월 29일 (수) 03:40 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 \(\ddot{x}+\omega^{2} x=0\)
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 \(1/2\omega, (1 +1/2)\omega, (2 +1/2)\omega,(3 +1/2)\omega,\cdots\) 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
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
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