View source for Coherent state (Quantum mechanics)

==Spin Coherent States==

An analogous story applies to other sets of coherent states, such as the spin coherent states,
defined for <math>\hbar=1</math> for a fixed spin value <math>s\in\{1/2,1,3/2,2,\ldots\}</math> by 

:<math>\label{eleven}
|\theta,\phi\rangle= e^{\textstyle-i\phi S_3}\,e^{\textstyle-i\theta S_2}\,|s;s\rangle\;, </math>


where the fiducial vector <math>|s;s\rangle</math> is the extremal weight vector of the operator
<math>S_3\ ,</math> i.e., one of the family of eigenvectors such that <math>S_3\,|m;s\rangle=m\,|m;s\rangle\ ,</math> <math>-s\leq m\leq s\ .</math> These coherent states are evidently continuous in their labels, and moreover

:<math>\label{twelve}
\int|\theta,\phi\rangle\langle\theta,\phi|\,
d\mu(\theta,\phi)=I_s\;,\;\;\;\;\;\;\;\;\;\;\; d\mu(\theta,\phi)=\frac{2s+1}{4\pi}\,\sin(\theta)\,d\theta\,d\phi\;,</math>


where <math>I_s</math> is the <math>(2s+1)\times(2s+1)</math> unit matrix.

These states lead to a coherent state representation of Hilbert space by <math>\psi(\theta,
\phi)=\langle\theta,\phi|\psi\rangle\ ,</math> and the propagator of a dynamical spin Hamiltonian 
is given by

:<math>\label{thirteen}
K(\theta'',\phi'',T;\theta',\phi',0)=\int e^{ i\int[i\langle\theta,\phi|
(d/dt)|\theta,\phi\rangle-\langle\theta,\phi|\textbf{H}|\theta,\phi\rangle]dt}\;D\mu(\theta.\phi)=\int e^{ i\int[s\cos(\theta){\dot\phi}-H(\theta,\phi)]dt}\,D\mu(\theta,\phi)\;.</math>