Update 4_ZPFs.md

src/4_ZPFs.md

@@ -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}
     $$