Cramer-Rao Inequality for Fisher Information and Zeros of the Wavefunction
| Main Author: | Francesco R. Ruggeri |
|---|---|
| Format: | info publication-preprint Journal |
| Terbitan: |
, 2019
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| Subjects: | |
| Online Access: |
https://zenodo.org/record/3235298 |
Daftar Isi:
- The Cramer-Rao inequality (which is based on a Cauchy-Schwarz result for inner products of vectors) links spatial variance to Fisher information. The inner products are integrals in this case, and it seems the entire bound state space is utilised and use is made of a vanishing wavefunction at endpoints. It seems, however, the wavefunction may vanish at other points if it is periodic, such as for an oscillator or a particle in a box. In such a case, it seems one could use Fisher information to analyze density humps and obtain information about the variance of x within such a hump, with the origin shifted to the center of the hump. In particular, as energy levels rise, the zeroes come close together and one may associate the envelope of quantum density W(x)W(x) (W(x)=wavefunction) with C/v(x) (v(x)= classical velocity). Thus, at very high energy levels, one may combine Fisher information with the correspondence principle to obtain information about spatial quantum variance in terms of classical quantities.