  Using a left multiplication defined on a right quaternionic Hilbert space,
linear self-adjoint momentum operators on a right quaternionic Hilbert space
are defined in complete analogy with their complex counterpart. With the aid of
the so-obtained position and momentum operators, we study the Heisenberg
uncertainty principle on the whole set of quaternions and on a quaternionic
slice, namely on a copy of the complex plane inside the quaternions. For the
quaternionic harmonic oscillator, the uncertainty relation is shown to saturate
on a neighborhood of the origin in the case we consider the whole set of
quaternions, while it is saturated on the whole slice in the case we take the
slice-wise approach. In analogy with the complex Weyl-Heisenberg Lie algebra,
Lie algebraic structures are developed for the quaternionic case. Finally, we
introduce a quaternionic displacement operator which is square integrable,
irreducible and unitary, and we study its properties.
