Harmonic oscillator in quantum mechanics

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

• Schrodinger equation

\(E \psi = -\frac{\hbar^2}{2m}{\partial^2 \psi \over \partial x^2} + V(x)\psi\)

\(V(x)=\frac{k}{2}x^2\)

 

 

 

path integral formulation

 

 

 

Groundstate correlation functions

 

 

 

 

 

related items

• free massless boson
• partition function in string theory

 

 

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)

 

 

 

expositions

• http://web.physik.rwth-aachen.de/~meden/vielteilchenneu/skriptka2.pdf
• http://www.odu.edu/~jdudek/lecture_notes/GradQM_Second_Semester/ProblemSet0.pdf