The number‐phase uncertainty products proposed by Carruthers and Nieto are studied to determine whether they are minimized by coherent states. It is found that coherent states do not minimize these products. States that do minimize some of the uncertainty products are constructed. Variational techniques for the study of arbitrary uncertainty products are developed.

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L.
Susskind
 and
J.
Glogower
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Physics
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1964
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2.
P.
Carruthers
 and
M. M.
Nieto
,
Phys. Rev. Letters
14
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387
(
1965
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3.
R. J.
Glauber
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Phys. Rev.
131
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4.
If |Ψ〉 is an eigenstate of one of the two operators, say X, then (ΔX)2 = 0, and (ΔY)2 necessarily diverges when A is a c number. Therefore the uncertainty product is of the indeterminate form 0⋅∞. We find in our subsequent analysis that such an indeterminate quantity can be sometimes evaluated; see Appendix A.
5.
Eigenstates of X and Y pose a special problem. For suppose we take |Ψ〉 to be an eigenstate of X, and assume that (ΔY)2 diverges so that the problem is nontrivial. Then Eq. (8) has the indeterminate form 0/0|Ψ〉+Ŷ2/∞|Ψ−2Ψ〉 = 0. Evidently an effective point of view is to ignore those solutions of (8) which are eigenstates of X and Y, and evaluate U separately with the eigenstates to determine whether these minimize U.
6.
In the direct method, the parameters λ and y need not be evaluated separately since their value is set by the form of Eq. (5a). Indeed the four conditions in (9b) are redundant since the form of Eq. (9a) assures that one relation between the parameters exists, viz.,
〈χ2〉−2α〈χ〉+α2a2+〈Y2〉−2β〈Y〉+β2b2 = 2
.
7.
G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, London, 1952), p. 294.
8.
Reference 7, p. 172.
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© 1968 The American Institute of Physics.
1968
The American Institute of Physics
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