Commit 7ad4049b authored by T. van der Sar's avatar T. van der Sar
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Update 4_ZPFs.md - more typos

parent 6cc51ef6
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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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