 ### Update 4_ZPFs.md - polish

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 ... ... @@ -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: $\alpha(t) = e^{-i \omega t} \alpha$. ... ...
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