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

2012년 2월 13일 (월) 07:53 판

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
\(\left[a , a^{\dagger} \right] = 1\)
\(\left[ H, a \right]= - \hbar \omega a\)
\(\left[ H, a^\dagger \right] = \hbar \omega a^\dagger\)

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.

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

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

Groundstate correlation functions

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-<h5>harmonic oscillator in classical mechanics</h5>
-* 고전역학에서의 조화진동자([http://statphys.springnote.com/pages/5695329 고전역학에서의 가적분성] 항목 참조)
-*  질량 m, frequency <math>\omega</math> 인 조화진동자<br>
-*  해밀토니안<br><math>H(p,q)=\frac{p^2}{2m}+\frac{m}{2}\omega^{2}q^2</math><br>
-*  해밀턴 방정식<br><math>\dot{q}=\partial H/\partial p=\frac{p}{m}</math><br><math>\dot{p}=-\partial H/\partial q=-m\omega^{2}q</math><br>
-*  운동방정식<br><math>\ddot{x}=-\omega^{2} x</math> 즉 <math>\ddot{x}+\omega^{2} x=0</math><br>
-<h5>quantum harmonic oscillator</h5>
-*  해밀토니안<br><math>H(P,X)=\frac{P^2}{2m}+\frac{m}{2}\omega^{2}X^2</math><br>
 <h5>creation and annhilation operators</h5>