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

2012년 2월 13일 (월) 08:20 판

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.

Groundstate correlation functions

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-<h5>creation and annhilation operators</h5>
+[[Heisenberg group and Heisenberg algebra]]
-*  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>
+[[Schrodinger equation]]
-*  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>
-*  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>
@@ 18번째 줄: / 17번째 줄: @@
-<h5>Schrodinger equation</h5>
-* [[Schrodinger equation]]
-<math>E \psi = -\frac{\hbar^2}{2m}{\partial^2 \psi \over \partial x^2} + V(x)\psi</math>
-<math>V(x)=\frac{k}{2}x^2</math>