  Attempts to formulate a quantum theory of gravitation are collectively known
as {\it quantum gravity}. Various approaches to quantum gravity such as string
theory and loop quantum gravity, as well as black hole physics and doubly
special relativity theories predict a minimum measurable length, or a maximum
observable momentum, and related modifications of the Heisenberg Uncertainty
Principle to a so-called generalized uncertainty principle (GUP). We have
proposed a GUP consistent with string theory, black hole physics and doubly
special relativity theories and have showed that this modifies all quantum
mechanical Hamiltonians. When applied to an elementary particle, it suggests
that the space that confines it must be quantized, and in fact that all
measurable lengths are quantized in units of a fundamental length (which can be
the Planck length). On the one hand, this may signal the breakdown of the
spacetime continuum picture near that scale, and on the other hand, it can
predict an upper bound on the quantum gravity parameter in the GUP, from
current observations. Furthermore, such fundamental discreteness of space may
have observable consequences at length scales much larger than the Planck
scale. Because this influences all the quantum Hamiltonians in an universal
way, it predicts quantum gravity corrections to various quantum phenomena.
Therefore, in the present work we compute these corrections to the Lamb shift,
simple harmonic oscillator, Landau levels, and the tunneling current in a
scanning tunneling microscope.
