  The subject of the present paper is an application of quantum probability to
$p$-adic objects. We give a quantum-probabilistic interpretation of the
spherical Hecke algebra for ${\rm PGL}_2(F)$, where $F$ is a $p$-adic field. As
a byproduct, we obtain a new proof of the Fourier inversion formula for ${\rm
PGL}_2(F)$.
