发表机构
University at Buffalo, State University of New York（纽约州立大学布法罗分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文从二进制序列计数原理出发，通过三条信息论公设推导出量子谐振子，证明干涉与玻恩规则源于计数对称性，并给出可检验的有限修正。
AI 中文摘要
量子谐振子（QHO）通常建立在包含希尔伯特空间、阶梯算子和玻恩规则的结构之上。本文从计数原理出发推导量子谐振子。我们采用三条信息论公设：信息由长度为 $n$ 的二进制序列承载；只有符号计数是可观测的，序列本身不可观测；转移测度必须精确计入寄存器的 $2^n$ 种构型容量。前两条公设将计数问题强制纳入Hamming/Delsarte结合方案，并赋予其在两种符号重新标记下的精确对称性。给定一个交换对称的双线性转移测度，我们证明了一个刚性定理：重新标记对称性与归一化共同强制产生驱动干涉的交变符号 $(-1)^t$。携带该符号的变量，即输入与输出序列之间的重叠度，等价于它们的Hamming距离 $d_{ab}$，是整个转移的一个隐藏量子数，仅由两个端点联合定义，单独任一端点均无法得知。因此，干涉作为对非局域、关系性变量 $d_{ab}$ 的交变符号记账而进入，而非作为独立的动力学要素。这种强制加权使转移概率正比于Krawtchouk多项式的平方，其差分方程具有精确等间距谱，并在 $n\to\infty$ 时收敛于量子谐振子。在固定激发下，与量子谐振子的偏差为 $\mathcal{O}(1/n)$，可在实验室实现的谐振子中检验，因为任何真实系统都具有有限信息容量。在此框架下，干涉、二次型（类玻恩规则）概率以及粒子-空穴对称的谐振子谱都是计数的结果；仅双线性测度形式是假设的（尽管有充分动机）而非推导所得。
英文摘要
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 $n\to\infty$. At fixed excitation, deviations from the QHO are $\mathcal{O}(1/n)$, 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.
Comments15 pages, 7 figures
