Update 4_ZPFs.md

src/4_ZPFs.md

@@ -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})\\