 ### Update 4_ZPFs.md - typo

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 ... ... @@ -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 $\alpha$. ... ...
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