Commit 6f5eced0 authored by T. van der Sar's avatar T. van der Sar
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Update 4_ZPFs.md

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......@@ -128,7 +128,7 @@ where $\alpha$ is complex. The coherent state has the following properties:
- The coherent state is *not* an eigenstate of the Hamiltonian. Therefore, generally, it will "move" (i.e., $\langle x \rangle$ and $\langle p \rangle$ oscillate in time).
- Instead, the coherent state is an eigenstate of the annihilation operator $\hat{a}$.
??? Hint
???+ note
Recall that the creation operator $\hat{a}^\dagger$ and the annihilation operator $\hat{a}$ have the following properties:
$$
\begin{align}
......@@ -137,11 +137,11 @@ where $\alpha$ is complex. The coherent state has the following properties:
\hat{a}^\dagger\hat{a}|n\rangle =& n|n\rangle.
\end{align}
$$
The operator $\hat{n}=\hat{a}^{\dagger}\hat{a}$ is called the number operator. It allows us to write the Hamiltonian as $\hat{H}=\hbar\omega(\hat{a}^\dagger\hat{a}+0.5)$. Comparing this to
The last operator, $\hat{n}=\hat{a}^{\dagger}\hat{a}$, is called the number operator. It allows us to write the Hamiltonian as $\hat{H}=\hbar\omega(\hat{a}^\dagger\hat{a}+0.5)$. Comparing this to
$$
\hat{H}=\frac{\hat{p}^2}{2m}+\frac{1}{2}m\omega_0^2\hat{x}^2,
$$
we can derive
we can express the position and momentum operators in terms of $\hat{a}$ and $\hat{a}^\dagger$:
$$
\begin{align}
\hat{x} = &x_\text{ZPF}(\hat{a}^\dagger+\hat{a})\\
......
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