Non-Complex Harmonization

On one hand 

$$\textbf{p}=\frac{1}{c^2}\int_{V}\textbf{S}d\tau,$$

on the other hand

$$\textbf{p}=\hbar\textbf{k}.$$

Not all waves are harmonic. So we must first find a way to generalize de Broglie's relation to all waves. One approach is to work with $$\Psi=e^{i(\textbf{k}\cdot\textbf{x}-\omega t)},$$ yielding $$\textbf{k}=-i\frac{\nabla\Psi}{\Psi}$$ on which I have written extensively. The disadvantage of this approach is that imaginary unit \(i\) shows up, which is not ideal. 

Another is to work with $$\Psi=\sin(\textbf{k}\cdot\textbf{x}-\omega t),$$ which yields

$$\boxed{\textbf{k}=\frac{\nabla\Psi}{\sqrt{1-\Psi^2}}}$$

Advantages

• No \(i\)
• Works for vector (and tensor) \(\Psi\)