 ### Update 4_ZPFs.md - more typos

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 ... ... @@ -73,7 +73,7 @@ $$This is looking a lot like a harmonic oscillator$$ E = m \omega_0^2 x^2 + \frac{p^2}{2 m} E = \frac{1}{2} m \omega_0^2 x^2 + \frac{p^2}{2 m} $$where the electric and magnetic fields play the roles of position and momentum, respectively. ... ... @@ -127,6 +127,7 @@ where \alpha is complex. The coherent state has the following properties: - The coherent state is *not* an eigenstate of the Hamiltonian. Therefore, generally, it will "move" (i.e., \langle x \rangle and \langle p \rangle oscillate in time). - Instead, the coherent state is an eigenstate of the annihilation operator \hat{a}. ??? Hint Recall: the creation operator \hat{a}^\dagger and the annihilation operator \hat{a} have the following properties: - \hat{a}|n\rangle = \sqrt{n}|n-1\rangle, and \hat{a}|0\rangle=0. ... ... @@ -134,7 +135,7 @@ where \alpha is complex. The coherent state has the following properties: - \hat{a}^\dagger\hat{a}|n\rangle = n|n\rangle. Therefore, \hat{a}^{\dagger}\hat{a} is called the number operator. The number operator enables us to write the harmonic oscillator Hamiltonian as \hat{H}=\hbar\omega(\hat{a}^\dagger\hat{a}+0.5). Comparing this to$$ \hat{H}=\hat{p}^2/2m+0.5m\omega_0^2\hat{x}^2, \hat{H}=\frac{\hat{p}^2}{2m}+\frac{1}{2}m\omega_0^2\hat{x}^2, $$we can derive that$$ ... ... @@ -144,6 +145,7 @@ where $\alpha$ is complex. The coherent state has the following properties: $$\hat{p} = i\frac{\hbar}{2 x_\text{ZPF}}(\hat{a}^\dagger-\hat{a})$$ - 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, coherent states evolve into new coherent states with an $\alpha$ that has the same amplitude but a different phase: $\alpha(t) = e^{-i \omega t} \alpha$. ... ...
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