Harmonic oscillator in quantum mechanics

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http://bomber0.myid.net/ (토론)님의 2011년 2월 8일 (화) 04:30 판
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introduction

 

 

 

harmonic oscillator in classical mechanics
  • 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

 

 

related items

 

 

encyclopedia

 

 

books

 

 

 

expositions

 

 

articles

 

 

 

question and answers(Math Overflow)

 

 

blogs

 

 

experts on the field

 

 

links
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