According to Nayak & Vishvanath’s paper ”Quantum walk on the line,” the probability of observing the particle doing the quantum walk at any give point n is

p(n, t) = ∥ψ_{L}(n, t)∥^{2} + ∥ψ_{R}(n, t)∥^{2}

where

ψ_{L} = \frac{1 + (−1)^{n+t}}{2} ∫_{−π}^{π} \frac{dk}{2π} (1 + \frac{cos k}{\sqrt{1 + cos^{2} k}}) exp\{−i(ω_{k}t + kn)\}

ψ_{R} = \frac{1 + (−1)^{n+t}}{2} ∫_{−π}^{π} \frac{dk}{2π} \frac{exp(ik)}{\sqrt{1 + cos^{2} k}}) exp\{−i(ω_{k}t + kn)\}