"양자 조화진동자"의 두 판 사이의 차이

2012년 2월 17일 (금) 05:52 판

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 \(\ddot{x}+\omega^{2} x=0\)

위치 연산자와 운동량 연산자
\([X,P] = X P - P X = i \hbar\)
\(\hat p = - i \hbar {\partial \over \partial x}\)
해밀토니안
\(\hat H(\hat p,\hat x) = \frac{{\hat p}^2}{2m} + \frac{1}{2} m \omega^2 {\hat x}^2\)
\(\hat H = \hbar \omega \left(a^{\dagger}a + 1/2\right)=\hbar \omega \left(a a^{\dagger }-\frac{1}{2}\right)\)
사다리 연산자(ladder operator)
\(a =\sqrt{m\omega \over 2\hbar} \left(\hat x + {i \over m \omega} \hat p \right)\)
\(a^{\dagger} =\sqrt{m \omega \over 2\hbar} \left( \hat x - {i \over m \omega} \hat p \right)\)
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
ground state energy of the oscillator
The lowest energy is not zero
It’s \(\omega/2\). This is called the.

@@ 41번째 줄: / 41번째 줄: @@
 * http://www.fisica.net/quantica/quantum_harmonic_oscillator_lecture.pdf
 * [[에르미트 다항식(Hermite polynomials)]]<br>
+<h5>energy  eigenstates</h5>
+*  Assume that Planck’s constant equals 1<br>
+*  a harmonic oscillator that vibrates with frequency <math>\omega</math> can have energy <math>\frac{\omega}{2}, (1 +\frac{1}{2})\omega, (2 +\frac{1}{2})\omega,(3 +\frac{1}{2})\omega,\cdots</math> in units where<br>
+*   ground state energy of the oscillator<br> The lowest energy is not zero<br>
+*  It’s <math>\omega/2</math>. This is called the.<br>