The Heisenberg group is one of the most simple non-Abelian Lie groups. The Lie algebra components (vector fields) X, Y, Z satisfy [X,Y] = Z. We recognise this relation from quantum mechanics, where the position and momentum operators satisfy this relation, or from signal processing, where it is satisfied by the operations of translating in frequency and translating in time. I have studied the Schrödinger equation formulated on the Heisenberg group, with the help of non-Abelian harmonic analysis. I will give some insight about how this differs from its Euclidean counterpart, and about some of the key techniques and ideas.