This paper examines the development and meaning of Heisenberg's notion of wave-particle equivalence and the way in which it differs from Bohr's more widely known notion of wave-particle complementarity. According to the statistical interpretation of the wave function, developed by Born and Pauli in 1926, the electron is treated as a particle, though it cannot be assigned a well-defined position and momentum at a given time. On the other hand, from the vantage point of quantum electrodynamics developed by Jordan, Klein and Wigner in 1927-1928, the electron is described as a quantized matter wave in three-dimensional space. Heisenberg brought these two empirically equivalent approaches together in his 1929 Chicago lectures. Whereas Bohr argued that it was necessary to use wave and particle descriptions alternatively in different experimental arrangements, Heisenberg insisted that one could interpret the quantum-mechanical equation of motion in terms of either a wave ontology or a particle ontology. Clarifying the differences between Bohr and Heisenberg provides a deeper insight into the divergent views which formed the so-called 'Copenhagen interpretation' of quantum mechanics.