# When solving a paraxial Helmholtz equation ... ? Amplitude vs Wave function

Given the paraxial Helmholtz-equation (PHHE). Why are the solutions formed by the amplitude function of a wave function, but not the wave function itself? Example, a form of a wave function is defined $$a(x) = A(x)e^{ikz}$$ with A(x) being the transversal profile/ amplitude function. Why does the explicit A(x) solves the PHHE and not the a(x). When deriving the gaussian beam, you also get a form of a wave function: $$\large a(x) = A_0 \frac{w_0}{w(z)}e^{\frac{-r^2}{w^2}}e^{\frac{-ikr^2}{2R} }e^{-i(kz-arctan(z/z_0))}$$. What term inside the wave function would solve the PHHE? Is there a way to imagine what is going on? Thank you

I'm not sure what is the par-axial Helmholtz equation. But consider the function $$a(x,z) = A(x) e^{i k z},$$ and consider the Laplace equation $$∆ a(x,z) = \nabla^2 a(x,z) =\frac{\partial^2 a(x,y)}{\partial x^2} + \frac{\partial^2 a(x,y)}{\partial z^2}= 0,$$ simply applying this we get $$\frac{\partial^2 A(x)}{\partial x^2} e^{i k z} - k^2 A(x) e^{ikz} = 0,$$ crossing out the exponential we get
$$\left(\frac{\partial^2 }{\partial x^2}-k^2\right)A(x) = 0.$$
Thus if $$a(x,z)$$ solves the Laplace equation and is in the form $$a(x,z) = A(x) e^{i k z}$$ then $$A(x)$$ solves the Helmholtz equation in the form given above.