  The existence of an observer independent minimum length scale can lead to the
modification of the Heisenberg uncertainty principle to the generalized
uncertainty principle. This in turn would be responsible for the modification
of the Hamiltonian describing a non-relativistic particle moving in the
presence of an arbitrary potential. In this work we carry out a path integral
formulation to compute the transition amplitude for this particle. The
formalism yields the action of such a particle in an arbitrary potential.
Interestingly, the action indicates that there is an upper bound to the
velocity that a particle can have which depends on the generalized uncertainty
principle parameter. We then compute explicitly the propagator of a free
particle and particle moving in a harmonic oscilltor potential using the path
integral representation of the transition amplitude. We observe that there
exists a curious connection between the transition amplitude of the free
particle in the generalized uncertainty priciple framework with the
corresponding result in noncommutative space found from the path integral
formulation in \cite{sgprl}. From the harmonic oscillator result for the
transition amplitude, we calculate the ground state energy of the harmonic
oscillator. The result shows that the ground state energy of the harmonic
oscillator in the framework of the Heisenberg uncertainty principle gets
augmented by the presence of the generalized uncertainty principle and also
depends on the mass of the particle. We also demonstrate that the result agrees
with that obatined using the operatorial approach.
