The concept of a wave equation exists in classical physics. For example, propagating electromagnetic radiation obeys a wave equation. Sound waves obey a different wave equation. Electromagnetic waves are an example of transverse waves and sound waves are known as longitudinal waves. However, the meaning of the term wave equation is quite different in quantum mechanics. A classical wave describes a propagating motion in which energy moves through a medium with both a spatial and time dependence. The quantum mechanical wave equation describes the probability of particles in space rather than their motion in a coordinate system. There is no time dependence needed in a quantum mechanical wave equation. We can introduce time dependence in the quantum mechanical wave equation, but that refers to some time-dependent change in the state of the system. The stationary wave eauation in quantum mechanics may be compared to an equation for a standing wave in classical physics.
We have seen that the momentum in the wave equation is p = h_bar k. Since h_bar = h/2 pi and k = 2 pi/lamda we can also rewrite the momentum that comes from the wave equation as p = h/lamda. This is the DeBroglie relation. Thus, we see that the quantum mechanical wave equation implies the wave-particle duality. It is built into the equaion. We do not need to add this concept, but rather it is fundamental to the concept of a wave equation in quantum mechanics..
