Heisenberg Inequalities for Wavelet States
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We prove a number of uncertainty results for wavelet states, the simplest one being that if a wavelet state is real-valued or, more generally, has zero expected momentum, then the Heisenberg uncertainty is at least 3/2 instead of the universal 1/2. For wavelet states having a very mild nth-order decay property, we establish a similar result for the uncertainty based on the nth powers of position and momentum, with a lower bound that grows rapidly with n. The proof is extremely elementary. Other Heisenberg inequalities are proven which involve the deviations about the origin of phase space rather than about the mean position of the wavelet in phase space, and the scaling generator plays an even more direct role than in the result mentioned above. The proof is still very elementary, combining the interscale orthogonality property with an iterated application of Rolle's Theorem. Naturally, the lower bounds are much greater for these deviations about the origin of phase space, but this yields consequences for how much "off-center" a wavelet must be if its uncertainty is approximately minimized. © 1997 Academic Press.
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