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\markright{(Pattern Recognition)}

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\noindent
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 

\begin{displaymath}
p(n, t) = \| \psi_L (n, t) \|^2 + \| \psi_R (n, t) \|^2
\end{displaymath}

\noindent
where

\begin{displaymath}
\psi_L = \frac{1+(-1)^{n+t}}{2} \int_{-\pi}^{\pi}{\frac{dk}{2 \pi}}
(1 + \frac{\cos k}{\sqrt{1 + \cos^2 k}}) \exp \{{-i(\omega_k t + kn)\}}
\end{displaymath}

\begin{displaymath}
\psi_R = \frac{1+(-1)^{n+t}}{2} \int_{-\pi}^{\pi}{\frac{dk}{2 \pi}}
\frac{\exp (i k)}{\sqrt{1 + \cos^2 k}}) \exp \{{-i(\omega_k t + kn)\}}
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