Commit 22dddfca authored by T. van der Sar's avatar T. van der Sar
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Update 4_ZPFs.md

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......@@ -132,7 +132,7 @@ where $\alpha$ is complex. The coherent state has the following properties:
Recall that the creation operator $\hat{a}^\dagger$ and the annihilation operator $\hat{a}$ have the following properties:
$$
\begin{align}
\hat{a}|n\rangle = & \sqrt{n}|n-1\rangle \, \text{and} \, \hat{a}|0\rangle=0, \\
\hat{a}|n\rangle = & \sqrt{n}|n-1\rangle \quad \text{and} \quad \hat{a}|0\rangle=0, \\
\hat{a}^\dagger|n\rangle = & \sqrt{n+1}|n+1\rangle,\\
\hat{a}^\dagger\hat{a}|n\rangle =& n|n\rangle.
\end{align}
......@@ -141,11 +141,11 @@ where $\alpha$ is complex. The coherent state has the following properties:
$$
\hat{H}=\frac{\hat{p}^2}{2m}+\frac{1}{2}m\omega_0^2\hat{x}^2,
$$
we can express the position and momentum operators in terms of $\hat{a}$ and $\hat{a}^\dagger$:
we can derive
$$
\begin{align}
\hat{x} = &x_\text{ZPF}(\hat{a}^\dagger+\hat{a})\\
\hat{p} = i\frac{\hbar}{2 x_\text{ZPF}}(\hat{a}^\dagger-\hat{a}).
\hat{p} = & i\frac{\hbar}{2 x_\text{ZPF}}(\hat{a}^\dagger-\hat{a}).
\end{align}
$$
......
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