 ### Update 4_ZPFs.md

parent 47c2c6e1
Pipeline #47329 passed with stages
in 6 minutes and 56 seconds
 ... ... @@ -128,7 +128,7 @@ 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 ???+ note Recall that the creation operator $\hat{a}^\dagger$ and the annihilation operator $\hat{a}$ have the following properties: \begin{align} ... ... @@ -137,11 +137,11 @@ where \alpha is complex. The coherent state has the following properties: \hat{a}^\dagger\hat{a}|n\rangle =& n|n\rangle. \end{align} 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 The last 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 $$\hat{H}=\frac{\hat{p}^2}{2m}+\frac{1}{2}m\omega_0^2\hat{x}^2,$$ we can derive we can express the position and momentum operators in terms of $\hat{a}$ and $\hat{a}^\dagger$:  \begin{align} \hat{x} = &x_\text{ZPF}(\hat{a}^\dagger+\hat{a})\\ ... ...
Supports Markdown
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!