Update 4_ZPFs.md - polish

src/4_ZPFs.md

@@ -129,23 +129,27 @@ where $\alpha$ is complex. The coherent state has the following properties:
 - Instead, the coherent state is an eigenstate of the annihilation operator $\hat{a}$.
 
 ??? Hint 
-    Recall: the creation operator $\hat{a}^\dagger$ and the annihilation operator $\hat{a}$ have the following properties:
-    - $\hat{a}|n\rangle = \sqrt{n}|n-1\rangle$, and $\hat{a}|0\rangle=0$.
-    - $\hat{a}^\dagger|n\rangle = \sqrt{n+1}|n+1\rangle$.
-    - $\hat{a}^\dagger\hat{a}|n\rangle =  n|n\rangle$. Therefore, $\hat{a}^{\dagger}\hat{a}$ is called the number operator.  
-    The number operator enables us to write the harmonic oscillator Hamiltonian as $\hat{H}=\hbar\omega(\hat{a}^\dagger\hat{a}+0.5)$. Comparing this to 
+    Recall that the creation operator $\hat{a}^\dagger$ and the annihilation operator $\hat{a}$ have the following properties:
     $$
-    \hat{H}=\frac{\hat{p}^2}{2m}+\frac{1}{2}m\omega_0^2\hat{x}^2,
+    \begin{align}
+    \hat{a}|n\rangle = & \sqrt{n}|n-1\rangle \, \text{and} \, \hat{a}|0\rangle=0, \\
+    \hat{a}^\dagger|n\rangle = & \sqrt{n+1}|n+1\rangle,\\
+    \hat{a}^\dagger\hat{a}|n\rangle =&  n|n\rangle.
+    \end{align}
     $$
-    we can derive that  
+    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 
     $$
-    \hat{x} = x_\text{ZPF}(\hat{a}^\dagger+\hat{a})
+    \hat{H}=\frac{\hat{p}^2}{2m}+\frac{1}{2}m\omega_0^2\hat{x}^2,
     $$
-    and
+    we can derive  
     $$
+    \begin{align}
+    \hat{x} = &x_\text{ZPF}(\hat{a}^\dagger+\hat{a})\\
     \hat{p} = i\frac{\hbar}{2 x_\text{ZPF}}(\hat{a}^\dagger-\hat{a}).
+    \end{align}
     $$
     
+    
 - There are an infinite number of possible coherent states, since $\alpha$ can vary continuously: $\alpha = |\alpha|e^{i \theta}$.
 - All coherent states are Heisenberg-limited minimum uncertainty wavepackets that satisfy $\sigma_x \sigma_p = \frac{\hbar}{2}$ (see exercise).
 - As a function of time, a coherent state evolves into a new coherent state with the same amplitude but a different phase: $\alpha(t) = e^{-i \omega t} \alpha$.