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Journal of Physics A Mathematical Nuclear and General

On the uncertainty relation in the coherent spin-state representation

L Kolodziejczyk, A Ryter

Published Jan 21, 1974• 6 citations
Publisher

Abstract

The product ( delta s x ) 2 ( delta s y ) 2 is computed where ( delta s x,y ) 2 =(s x,y 2 )-(s x,y ) 2 ; averaging is performed in the coherent spin-state representation given by Radcliffe (1971). After applying the Holstein- Primakoff transformation s - =(2s) 1 2/a Dagger , s + =(2s) 1 2/a and mu = alpha /(2s) 1 2/, and putting s to infinity one proceeds from Radcliffe space into the Glauber space. After this procedure the product ( delta s x ) 2 ( delta s y ) 2 becomes ( delta x) 2 ( delta p) 2 . Using the Jackiw equation it is shown that the function mod mu =0> is the only one which minimizes the uncertainty product, for every s.

DOI
10.1088/0305-4470/7/2/007

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The product ( delta s x ) 2 ( delta s y ) 2 is computed where ( delta s x,y ) 2 =(s x,y 2 )-(s x,y ) 2 ; averaging is performed in the coherent spin-state representation given by Radcliffe (1971). After applying the Holstein- Primakoff transformation s - =(2s) 1 2/a Dagger , s + =(2s) 1 2/a and mu = alpha /(2s) 1 2/, and putting s to infinity one proceeds from Radcliffe space into the Glauber space. After this procedure the product ( delta s x ) 2 ( delta s y ) 2 becomes ( delta x) 2 ( delta p) 2 . Using the Jackiw equation it is shown that the function mod mu =0> is the only one which minimizes the uncertainty product, for every s.
The author shows that the measure of angular momentum uncertainty is the invariant ( Delta J) 2 identical to (J 2 )-(J) 2 , rather than sums of products like Delta J 1 Delta J 2 , and that the critical states which minimise Delta J are eigenstates of maximum weight of J.n. They also determine the critical states for the associated groups E(2), O(2,1), O(4) and E(3), and construct coherent O(3) spin states mod z) which are superpositions of normal spin eigenvectors and which tend to the classical limit as mod z mod to infinity .
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.
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.