  Various theories of Quantum Gravity argue that near the Planck scale, the
Heisenberg Uncertainty Principle should be replaced by the so called
Generalized Uncertainty Principle (GUP). We show that the GUP gives rise to two
additional terms in any quantum mechanical Hamiltonian, proportional to \beta
p^4 and \beta^2 p^6 respectively, where \beta \sim 1/(M_{Pl}c)^2 is the GUP
parameter. These terms become important at or above the Planck energy.
Considering only the first of these, and treating it as a perturbation, we show
that the GUP affects the Lamb shift, Landau levels, reflection and transmission
coefficients of a potential step and potential barrier, and the current in a
Scanning Tunnel Microscope (STM). Although these are too small to be measurable
at present, we speculate on the possibility of extracting measurable
predictions in the future.
