I have a little question about equation for creation/annihilation operators.

Usually we obtain time evolution equation for these operators from Heisenberg equation. for example:

$$\frac{da_l}{dt} = -j\omega_l a_l$$

But in some papers I saw the case of interaction between travelling waves, where time was replaced by coordinate and frequency by wave vector, e.g.

$$\frac{da_l}{dz} = -j k_l a_l$$

Intuitively, this expression is clear. However since we initially assumed that operators are dependent on time, I would like to know how to implement this transition strictly.



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