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Journal of Physics A Mathematical and General

Rotations and intelligence of coherent spin states

G. Vetri

Published Jun 1, 1975• 8 citations
Publisher

Abstract

The minimum uncertainty relation which characterizes a coherent spin state is shown to be invariant under some special rotations and a model to interpret this result is given. 'Intelligence' of coherent states is discussed in the light of this interpretation.
DOI
10.1088/0305-4470/8/6/002

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The minimization of the uncertainty relation $2\ensuremath{\Delta}x\ensuremath{\Delta}p\ensuremath{\ge}\ensuremath{\hbar}$ characterizes the coherent states. The spincoherent states of Radcliffe minimize the uncertainty relation $2\ensuremath{\Delta}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{J}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{u}})\ensuremath{\Delta}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{J}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}})\ensuremath{\ge}|〈\stackrel{\ensuremath{\rightarrow}}{\mathrm{J}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{w}}〉|$ (where $\stackrel{\ensuremath{\rightarrow}}{\mathrm{u}}$, $\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}$, $\stackrel{\ensuremath{\rightarrow}}{\mathrm{w}}$ form an arbitrary orthonormal basis), but this property does not characterize them. Here, we analyze that situation and suggest an alternative to restore the parallelism.
Minimum uncertainty states of the conventional Heisenberg uncertainty relation have been extensively studied and are often regarded as the most classical quantum states from the perspective of uncertainty, providing valuable insight into the nature of quantumness and its potential applications. In this work, we investigate the minimum uncertainty states associated with an information-theoretic refinement of the Heisenberg uncertainty relation in general spin systems. Using two different approaches, the matrix formulation and the Wick symbol representation, we derive explicit expressions for the states that saturate the uncertainty bound. We show that spin coherent states indeed achieve minimum uncertainty, consistent with their conventional identification as the classical states of spin systems. Moreover, we also identify additional classes of minimum uncertainty states beyond the coherent family. Finally, we compare the spin-system results with the previously studied bosonic case and elucidate the origin of the differences between the two settings.
The intelligent spin states are defined as those states which satisfy the Heisenberg equality for the spin operators: Delta J x 2 Delta J y 2 = mod (J z ) mod 2 . The 2j+1 states which behave intelligently in each angular momentum space of spin j are found explicitly. For this purpose the Radcliffe states are used, it is shown that only the real and the pure imaginary Radcliffe states are intelligent. These intelligent states also satisfy the quartic consistency condition. The result, however, does not disagree in principle with the recent claim of Kolodziejczyk and Ryter (1974) that mod mu )= mod 0) is the only state which minimizes the uncertainty product because minimum uncertainty does not necessarily imply intelligence.