  This paper is inspired by Wang, Wang and Zhang's work [ Observability and
unique continuation inequalities for the Schr\"odinger equation. J. Eur. Math.
Soc. 21, 3513--3572 (2019)], where they present several observability and
unique continuation inequalities for the free Schr\"{o}dinger equation in
$\mathbb{R}^{n}$. We extend all such observability and unique continuation
inequalities for the Schr\"{o}dinger equations on half-line with inverse-square
potentials. Technically, the proofs essentially rely on the representation of
the solution, a Nazarov type uncertainty principle for the Hankel transform and
an interpolation inequality for functions whose Hankel transform have compact
support.
