Editing Quantum harmonic oscillator (section)

===Coherent states===

{{main|Coherent state}}

[[File:Coherent state gif.gif|thumb|right|450px|Coherent state dynamics for <math>\alpha = \sqrt{10}</math>, in units of the harmonic oscillator length <math>x_0=\sqrt{\hbar/m\omega}</math>, showing the probability density <math>|\psi(x,t)|^2</math> and the quantum phase (color).]]

The [[Coherent states#The wavefunction of a coherent state|coherent states]] (also known as Glauber states) of the harmonic oscillator are special nondispersive [[wave packet]]s, with minimum uncertainty {{math|1=''σ<sub>x</sub>'' ''σ<sub>p</sub>'' = {{frac|''ℏ''|2}}}}, whose [[observable]]s' [[Expectation value (quantum mechanics)|expectation values]] evolve like a classical system.  They are eigenvectors of the annihilation operator, ''not'' the Hamiltonian, and form an [[Overcompleteness|overcomplete]] basis which consequentially lacks orthogonality.{{sfnp|Zwiebach|2022|pp=481-492}}

The coherent states are indexed by <math>\alpha \in \mathbb{C}</math> and expressed in the {{math|{{braket|ket|''n''}}}} basis as

<math display="block">|\alpha\rangle = \sum_{n=0}^\infty |n\rangle \langle n | \alpha \rangle = e^{-\frac{1}{2} |\alpha|^2} \sum_{n=0}^\infty\frac{\alpha^n}{\sqrt{n!}} |n\rangle = e^{-\frac{1}{2} |\alpha|^2} e^{\alpha a^\dagger} e^{-{\alpha^* a}} |0\rangle.</math>

Since coherent states are not energy eigenstates, their time evolution is not a simple shift in wavefunction phase. The time-evolved states are, however, also coherent states but with phase-shifting parameter {{mvar|α}} instead: <math>\alpha(t) = \alpha(0) e^{-i\omega t} = \alpha_0 e^{-i\omega t}</math>.<math display="block">|\alpha(t)\rangle = \sum_{n=0}^\infty e^{-i\left(n+\frac{1}{2}\right) \omega t}|n\rangle \langle n | \alpha \rangle =  e^{\frac{-i\omega t}{2}}e^{-\frac{1}{2} |\alpha|^2} \sum_{n=0}^\infty\frac{(\alpha e^{-i\omega t})^n}{\sqrt{n!}} |n\rangle = e^{-\frac{i\omega t}{2}}|\alpha e^{-i\omega t}\rangle </math>

Because <math>a \left| 0 \right\rangle = 0 </math> and via the Kermack-McCrae identity, the last form is equivalent to a [[Unitary operator|unitary]] [[displacement operator]] acting on the ground state: <math>|\alpha\rangle=e^{\alpha \hat a^\dagger - \alpha^*\hat a}|0\rangle = D(\alpha)|0\rangle</math>. Calculating the expectation values:

<math>\langle \hat{x} \rangle_{\alpha(t)} = \sqrt{\frac{2\hbar}{m\omega}}|\alpha_0|\cos{(\omega t - \phi)} </math>

<math>\langle \hat{p} \rangle_{\alpha(t)} = -\sqrt{2m\hbar \omega}|\alpha_0|\sin{(\omega t - \phi)} </math>

where <math>\phi </math> is the phase contributed by complex {{mvar|α}}. These equations confirm the oscillating behavior of the particle. 

The uncertainties calculated using the numeric method are:

<math>\sigma_x(t)=\sqrt{\frac{\hbar}{2m\omega}}  </math>

<math>\sigma_p(t) = \sqrt{\frac{m\hbar\omega}{2}} </math>

which gives <math display="inline">\sigma_x(t)\sigma_p(t) = \frac{\hbar}{2}  </math>. Since the only wavefunction that can have lowest position-momentum uncertainty, <math display="inline">\frac{\hbar}{2}  </math>, is a gaussian wavefunction, and since the coherent state wavefunction has minimum position-momentum uncertainty, we note that the general gaussian wavefunction in quantum mechanics has the form:<math display="block">\psi_\alpha(x')= \left(\frac{m\omega}{\pi\hbar}\right)^{\frac{1}{4}} e^{\frac{i}{\hbar} \langle\hat{p}\rangle_\alpha (x' - \frac{\langle\hat{x}\rangle_\alpha}{2}) - \frac{m\omega}{2\hbar}(x' - \langle\hat{x}\rangle_\alpha)^2} .</math>Substituting the expectation values as a function of time, gives the required time varying wavefunction.<br />

The probability of each energy eigenstates can be calculated to find the energy distribution of the wavefunction:

<math>P(E_n)=|\langle n | \alpha \rangle|^2 =  \frac{e^{-|\alpha|^2}|\alpha|^{2n}}{n!}</math>

which corresponds to [[Poisson distribution]].