Update 4_ZPFs.md - typo

src/4_ZPFs.md

@@ -139,7 +139,7 @@ Using the fact that $\alpha(t) = e^{-i \omega t} \alpha(0)$, we can see that the
 
 In quantum optics, $\hat{x}$ and $\hat{p}$ are called quadratures.
 
-The photon number operator $\hat{n}$ is $\hat{a}^\dagger \hat{a}$ (i.e. $\langle n | \hat{n} | n \rangle = n$), and the Hamiltonian then becomes $\hat{H} = (\frac{1}{2} + \hat{a}^\dagger \hat{a})$.  Using this, we find that
+The photon number operator $\hat{n}$ is $\hat{a}^\dagger \hat{a}$ (i.e. $\langle n | \hat{n} | n \rangle = n$), and the Hamiltonian then becomes $\hat{H} = \hbar \omega (\frac{1}{2} + \hat{a}^\dagger \hat{a})$.  Using this, we find that
 
 $$
 \langle \alpha | \hat{n} | \alpha \rangle = |\alpha|^2 = \bar{n}
@@ -154,7 +154,7 @@ Coherent states have maximal "classical" energy, and even though they are (infin
 
 - The expectation values of observables of stationary states do not depend on time. 
 - However, there are quantum fluctuations. These are related to the spread/curvature of the wavefunction.
-- The coherent state is not a stationary state. Its time dynamics resembles that of a classical particle. 
+- The coherent state is not a stationary state. Its time dynamics resemble that of a classical particle. 
 - The coherent state is an eigenstate of the annihilation operator.
 - The expectation values for position and momentum of the coherent state $|\alpha\rangle$ can be expressed in terms of $\alpha$.