2026/07/24 by Chien-Wen Hwang
#quant-ph #hep-ph #math-ph #math.MP
We introduce a discrete difference operator Dk 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± 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 (a2+b2)=2n-1, establishing probability conservation in the Krawtchouk formulation. The operator Dk acts as a discrete Hermite polynomial generator: the ratios hm = (n choose k)-1 Dkm (n choose k) admit explicit closed forms and converge to Hermite polynomials in the continuous limit. At the discrete level, Dk connects successive Krawtchouk matrices and acts as a coherence generator and a raising operator. The hm 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 hm (for x < 0), and the ballistic O(n2) scaling of the variance emerges from the superposition of all excited states. Numerical illustrations for n=6 and n=10 corroborate the analytical results.