Editing Coherent state (section)

== Mathematical features of the canonical coherent states ==
The canonical coherent states described so far have three properties that are mutually equivalent, since each of them completely specifies the state <math>|\alpha\rangle</math>, namely,
# They are eigenvectors of the [[annihilation operator]]: &nbsp; <math> \hat{a}|\alpha\rangle=\alpha|\alpha\rangle \,</math>.
# They are obtained from the vacuum by application of a unitary [[displacement operator]]: &nbsp;<math>|\alpha\rangle=e^{\alpha \hat a^\dagger - \alpha^*\hat a}|0\rangle  = D(\alpha)|0\rangle\,</math>.
# They are states of (balanced) minimal uncertainty: &nbsp; <math>\Delta X = \Delta P= \sqrt{\frac{\hbar}{2}}\,</math>.

Each of these properties may lead to generalizations, in general different from each other (see the article "[[Coherent states in mathematical physics]]" for some of these). We emphasize that coherent states have mathematical features  that are very different from those of a [[Fock state]]; for instance, two different coherent states are not orthogonal,
:<math>\langle\beta|\alpha\rangle=e^{-{1\over2}(|\beta|^2+|\alpha|^2-2\beta^*\alpha)}\neq\delta(\alpha-\beta)</math>
(linked to the fact that they are eigenvectors of the non-self-adjoint annihilation operator {{math|''â''}}).

Thus, if the oscillator is in the quantum state <math>|\alpha \rangle</math> it is also with nonzero probability in the other quantum state 
<math>|\beta \rangle</math> (but the farther apart the states are situated in phase space, the lower the probability is). However, since they obey a closure relation, any state can be decomposed on the set of coherent states. They hence form an '''''overcomplete basis''''', in which one can diagonally decompose any state. This is the premise for the [[Glauber–Sudarshan P representation]].

This closure relation can be expressed by the resolution of the identity operator {{mvar|I}} in the [[vector space]] of quantum states,
:<math>\frac{1}{\pi} \int |\alpha\rangle\langle\alpha| d^2\alpha = I
\qquad d^2\alpha \equiv d\Re(\alpha) \, d\Im(\alpha)  ~.</math>
This resolution of the identity is intimately connected to the [[Segal–Bargmann space#The Segal.E2.80.93Bargmann transform|Segal–Bargmann transform]].

Another peculiarity is that <math>\hat a^\dagger </math> has no eigenket (while {{math|''â''}} has no eigenbra). The following equality is the closest formal substitute, and turns out to be useful for technical computations,<ref>{{cite book |last1=Scully |first1=Marlan O. |last2=Zubairy |first2=M. Suhail |title=Quantum Optics |date=1997 |publisher=Cambridge University Press |location=Cambridge, UK |isbn=9780521435956 |page=67}}</ref>
:<math>
a^{\dagger}|\alpha\rangle \langle \alpha |=\left({\partial\over\partial\alpha}+\alpha^*\right)|\alpha\rangle \langle \alpha | ~.
</math>
This last state is known as an "Agarwal state" or photon-added coherent state and denoted as <math>|\alpha,1\rangle.</math>

Normalized Agarwal states of order {{mvar|n}} can be expressed as 
<math>|\alpha,n\rangle=[{\hat{a}^{\dagger}]}^n|\alpha\rangle / \| [{\hat{a}^{\dagger}]}^n|\alpha\rangle \|      ~.</math><ref>{{Cite journal|last1=Agarwal|first1=G. S.|last2=Tara|first2=K.|date=1991-01-01|title=Nonclassical properties of states generated by the excitations on a coherent state|journal=Physical Review A|volume=43|issue=1|pages=492–497|doi=10.1103/PhysRevA.43.492|pmid=9904801|bibcode=1991PhRvA..43..492A}}</ref>

The above resolution of the identity may be derived (restricting to one spatial dimension for simplicity) by taking matrix elements between eigenstates of position, <math> \langle x | \cdots | y \rangle </math>, on both sides of the equation. On the right-hand side, this immediately gives {{math| ''δ(x-y)''}}.  On the left-hand side, the same is obtained by inserting 
:<math>
   \psi^\alpha(x,t) = \langle x | \alpha(t)\rangle 
</math>
from the previous section (time is arbitrary), then integrating over  <math> \Im (\alpha) </math> using the [[Dirac delta function#Fourier transform|Fourier representation of the delta function]], and then performing a [[Gaussian integral]] over <math> \Re (\alpha) </math>.

In particular, the Gaussian Schrödinger wave-packet state follows from the explicit value 
:<math>\langle x | \alpha\rangle= \frac {1} {\pi^{1/4}}{e^{-\frac{1}{2}{(x-\sqrt{2}\Re(\alpha))^2} +ix \sqrt{2} \Im (\alpha)-i\Re(\alpha)\Im(\alpha)}}                    ~.</math>

The resolution of the identity may also be expressed in terms of particle position and momentum. For each coordinate dimension (using an adapted notation with new meaning for <math>x</math>), 
:<math> 
 |\alpha\rangle \equiv |x,p\rangle \qquad \qquad 
 x \equiv \langle \hat{x} \rangle \qquad\qquad  p \equiv \langle \hat{p} \rangle
</math>
the closure relation of coherent states reads
:<math>
 I = \int |x,p\rangle \, \langle x,p| ~ \frac{\mathrm{d}x\,\mathrm{d}p}{2\pi\hbar}   ~.
</math>
This can be inserted in any quantum-mechanical expectation value, relating it to some quasi-classical phase-space integral and explaining, in particular, the origin of normalisation factors <math> (2\pi\hbar)^{-1} </math> for classical [[partition function (statistical mechanics)|partition functions]], consistent with quantum 
mechanics.

In addition to being an exact eigenstate of annihilation operators, a coherent state is 
an ''approximate'' common eigenstate of particle position and momentum. Restricting to 
one dimension again, 
:<math>
    \hat{x} |x,p\rangle \approx x |x,p\rangle \qquad \qquad 
    \hat{p} |x,p\rangle \approx p |x,p\rangle 
</math>
The error in these approximations is measured by the [[uncertainty principle|uncertainties]] 
of position and momentum,
:<math>
    \langle x, p | \left(\hat{x} - x \right)^2 |x,p\rangle = \left(\Delta x\right)^2
    \qquad \qquad
    \langle x, p | \left(\hat{p} - p \right)^2 |x,p\rangle = \left(\Delta p\right)^2  ~.
</math>