Commit 47c2c6e1 authored by T. van der Sar's avatar T. van der Sar
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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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