AI 中文总结
研究用离散差分算子\(D_k\)研究一维量子随机游走，得到概率幅表达式，证明其满足帕塞瓦尔恒等式，\(D_k\)还充当离散埃尔米特多项式生成器，量化了量子干涉程度，数值例证证实分析结果。
AI 中文摘要
我们引入离散差分算子\(D_k\)来研究具有哈达玛硬币的一维量子随机游走。得到了概率幅\(a(n,k)\)和\(b(n,k)\)的显式组合表达式，它们编码了最终步方向并带有交替符号，反映向左步的合并。去除这些符号和硬币状态差异可恢复经典二项分布。对称和反对称组合\(a\pm b\)与克劳特楚克矩阵的对角和次对角元素一致。利用克劳特楚克矩阵元素间的交叉恒等式，通过归纳证明了幅满足帕塞瓦尔恒等式\(\sum(a^2 + b^2)=2^n - 1\)，在克劳特楚克公式中建立了概率守恒。算子\(D_k\)充当离散埃尔米特多项式生成器，比率\(h_m = \binom{n}{k}^{-1}D_k^m\binom{n}{k}\)有显式封闭形式并在连续极限中收敛到埃尔米特多项式。在离散层面，\(D_k\)连接连续的克劳特楚克矩阵，充当相干生成器和提升算子。\(h_m\)量化了分布两半上位置相关的量子干涉程度，方差的弹道\(O(n^2)\)缩放源于所有激发态的叠加。\(n = 6\)和\(n = 10\)的数值例证证实了分析结果。
英文摘要
We introduce a discrete difference operator D_k to study the one-dimensional quantum random walk (QRW) with the Hadamard coin. Explicit combinatorial expressions are obtained for the probability amplitudes a(n,k) and b(n,k), which encode the final step direction and carry alternating signs that reflect the merging of leftward steps. Removing these signs and the coin-state distinction recovers the classical binomial distribution. The symmetric and antisymmetric combinations $a\pm b$ are shown to coincide with diagonal and sub-diagonal entries of the Krawtchouk matrix. Using cross identities among Krawtchouk matrix elements, we prove by induction that the amplitudes satisfy a Parseval identity sum (a^2+b^2)=2^n-1, establishing probability conservation in the Krawtchouk formulation. The operator D_k acts as a discrete Hermite polynomial generator: the ratios h_m = (n choose k)^{-1} D_k^m (n choose k) admit explicit closed forms and converge to Hermite polynomials in the continuous limit. At the discrete level, D_k connects successive Krawtchouk matrices and acts as a coherence generator and a raising operator. The h_m quantify the position-dependent degree of quantum interference on both halves of the distribution, either individually (for x >= 0) or through an inverse-Pascal combination of several h_m (for x < 0), and the ballistic O(n^2) scaling of the variance emerges from the superposition of all excited states. Numerical illustrations for n=6 and n=10 corroborate the analytical results.
Comments24 pages, 1 figure
