  In this paper, we study the properties of lackadaisical quantum walks on a
line. This model is first proposed in~\cite{wong2015grover} as a quantum
analogue of lazy random walks where each vertex is attached $\tau$ self-loops.
We derive an analytic expression for the localization probability of the walker
at the origin after infinite steps, and obtain the peak velocities of the
walker. We also calculate rigorously the wave function of the walker starting
from the origin and obtain a long time approximation for the entire probability
density function. As an application of the density function, we prove that
lackadaisical quantum walks spread ballistically for arbitrary $\tau$, and give
an analytic solution for the variance of the walker's probability distribution.
