The stochastic quantization problem of Brownian particle motion that follows Kramers equation is
investigated in one-dimension using invariant related unitary transformation method. The explicit
Hamiltonians for Langevin equation are constructed separately for the systems that have several different
types of the external forces. In the case which does not have external forces, the spectrum of Schro?dinger
solution is continuous since the particles are unbound in a finite region. However, when a external force
is given by the (time-dependent) harmonic potential, the particles are bound inside the potential leading
the spectrum of quantum wave functions to be discrete. The bound?unbound transitions are important
since it is connected to metal?insulator transitions that can be achievable for certain compound
semiconductors by increasing the doping concentrations at low temperatures. Additionally, when the
Hamiltonian involves a higher order term of x as well as harmonic potential term, we have executed the
ordinary perturbation expansion in order to obtain the approximate quantum solutions.