  We construct families of squeezed quantum states on an interval (depending on
boundary conditions, we interpret the interval as a circle or as the infinite
square potential well) and obtain estimates of position and momentum
dispersions for these states. A particular attention is paid to the possibility
of proper localization of a particle in nanoscale space domains. One of the
constructed family of squeezed states is based on the theta function. It is a
generalization of the known coherent and squeezed states on the circle. Also we
construct a family of squeezed states based on truncated Gaussian functions and
a family of wave packets based on the discretization of an arbitrary continuous
momentum probability distribution.
  The problem of finiteness of the energy dispersion for the squeezed states in
the infinite well is discussed. Finally, we perform the limit of large interval
length and the semiclassical limit.
  As a supplementary general result, we show that an arbitrary physical
quantity has a finite dispersion if and only if the wave function of a quantum
system belongs to the domain of the corresponding self-adjoint operator. This
can be regarded as a physical meaning of the domain of a self-adjoint operator.
