Commit 373c0160 by T. van der Sar

### Update 4_ZPFs.md

 ... ... @@ -111,7 +111,7 @@ The first term is energy associated with classical motion, and the second comes So if all of the oscillator eigenstates are stationary, how can we get them to "move"? Suppose we prepare a superposition of two such states, like $|\Psi(t=0)\rangle=\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$. This will evolve in time as $$|\Psi(t) \frac{1}{\sqrt{2}}(e^{-iE_0t/\hbar}|0\rangle + e^{-iE_1t/\hbar}|1\rangle). |\Psi(t)\rangle= \frac{1}{\sqrt{2}}(e^{-iE_0t/\hbar}|0\rangle + e^{-iE_1t/\hbar}|1\rangle).$$ The relative phase between the two states will cause the observable $\langle x(t) \rangle$ to oscillate in time. ... ... @@ -143,15 +143,15 @@ where $\alpha$ is complex. The coherent state has the following properties: $$and$$ \hat{p} = i\frac{\hbar}{2 x_\text{ZPF}}(\hat{a}^\dagger-\hat{a}) \hat{p} = i\frac{\hbar}{2 x_\text{ZPF}}(\hat{a}^\dagger-\hat{a}). $$- There are an infinite number of possible coherent states since \alpha can vary continuously: \alpha = |\alpha|e^{i \theta}. - There are an infinite number of possible coherent states, since \alpha can vary continuously: \alpha = |\alpha|e^{i \theta}. - All coherent states are Heisenberg-limited minimum uncertainty wavepackets that satisfy \sigma_x \sigma_p = \frac{\hbar}{2} (see exercise). - As a function of time, coherent states evolve into new coherent states with an \alpha that has the same amplitude but a different phase: \alpha(t) = e^{-i \omega t} \alpha. - The ground state |0\rangle is also a coherent state with \alpha = 0 and \langle x \rangle = \langle p \rangle = 0. - As a function of time, a coherent state evolves into a new coherent state with the same amplitude but a different phase: \alpha(t) = e^{-i \omega t} \alpha. - The ground state |0\rangle is also a coherent state, with \alpha = 0 and \langle x \rangle = \langle p \rangle = 0. A coherent state \alpha has the following \langle x \rangle and \langle p \rangle: By expressing the operators for position and momentum in terms of creation and annihilation operators, it follows that a coherent state \alpha has:$$ \begin{align} ... ...