The quantum harmonic oscillator from binary sequences
Abstract
The quantum harmonic oscillator (QHO) is normally built on a structure that contains a Hilbert space, ladder operators, and a Born rule. Here we derive it from counting. We adopt three information-theoretic postulates: information is carried by binary sequences of length n; only symbol counts, not the sequences themselves, are observable; and the transition measure must account for exactly the 2n-configuration capacity of the register. The first two postulates force the counting problem into the Hamming/Delsarte association scheme and endow it with an exact symmetry under relabeling of the two symbols. Given an exchange-symmetric bilinear transition measure we prove a rigidity theorem: relabeling symmetry and normalization together force the alternating sign (−1)t which drives interference. The variable that carries the sign, i.e. the overlap between input and output sequences, or equivalently their Hamming distance dab, is a hidden quantum number of the transition as a whole, defined only by the two endpoints jointly and unknowable from either alone. Interference thus enters as alternating-sign bookkeeping over a nonlocal, relational variable dab, rather than as a separate dynamical ingredient. This forced weighting makes the transition probability proportional to the square of a Krawtchouk polynomial, whose difference equation has an exactly equally spaced spectrum and which converges to the QHO as n→∞. At fixed excitation, deviations from the QHO are 𝒪(1/n), and are testable in laboratory realizations of the oscillator, since any real system has finite information capacity. In this framework, interference, quadratic (Born-rule-like) probabilities, and the particle-hole symmetric oscillator spectrum are consequences of counting; only the bilinear form of the measure is assumed (though well motivated) rather than derived.
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BibTeX
@misc{powers2026quantum,
title = {{The quantum harmonic oscillator from binary sequences}},
author = {Samuel Powers and Dino Skrgic and Dejan Stojkovic},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10568},
eprint = {2609.10568},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10568},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The quantum harmonic oscillator from binary sequences
AU - Samuel Powers
AU - Dino Skrgic
AU - Dejan Stojkovic
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.10568
UR - https://arxiv.org/abs/2609.10568
AB - The quantum harmonic oscillator (QHO) is normally built on a structure that contains a Hilbert space, ladder operators, and a Born rule. Here we derive it from counting. We adopt three information-theoretic postulates: information is carried by binary sequences of length n; only symbol counts, not the sequences themselves, are observable; and the transition measure must account for exactly the 2^n-configuration capacity of the register. The first two postulates force the counting problem into the Hamming/Delsarte association scheme and endow it with an exact symmetry under relabeling of the two symbols. Given an exchange-symmetric bilinear transition measure we prove a rigidity theorem: relabeling symmetry and normalization together force the alternating sign (−1)^t which drives interference. The variable that carries the sign, i.e. the overlap between input and output sequences, or equivalently their Hamming distance d_ab, is a hidden quantum number of the transition as a whole, defined only by the two endpoints jointly and unknowable from either alone. Interference thus enters as alternating-sign bookkeeping over a nonlocal, relational variable d_ab, rather than as a separate dynamical ingredient. This forced weighting makes the transition probability proportional to the square of a Krawtchouk polynomial, whose difference equation has an exactly equally spaced spectrum and which converges to the QHO as n→∞. At fixed excitation, deviations from the QHO are 𝒪(1/n), and are testable in laboratory realizations of the oscillator, since any real system has finite information capacity. In this framework, interference, quadratic (Born-rule-like) probabilities, and the particle-hole symmetric oscillator spectrum are consequences of counting; only the bilinear form of the measure is assumed (though well motivated) rather than derived.
ER -