  We construct a quantum mechanics based on the hypothesis of existing compact
extra dimensions for a particle that wants to detect it. By introducing a
probability function, we express the transition of particle to the extra 2d
window. The general properties of this function has been examined and a length
scale for occurrence of particle to extra window is given. By a diverse view
point we consider that, the new length scale plays another quantum criteria for
another quantization, beside the Planck constant. Canonical quantization of
second class constrained systems, is our method for constructing the desired
quantum mechanics, in which in it the probability function enters in the
structure of second class constraints. This import the phenomena of extra
dimension to the 3d quantum mechanics, effectively. Some aspects of this
effective double quantum theory are mentioned, which one may investigate them
more focused to experience extra dimension quantum mechanically in contrast to
field theoretic sights. Specially, we try to make solutions for wave function
and spectrum of the free particle, by Frobenius prescription for solving linear
differential equations. In this context, the length scale of extra dimension
characterizes the singularity of the wave equation at the boundary which tiny
extra window connected to 3d space.
