Why is the modulus of a function taken before it is squared?

Question

I am a studying quantum mechanics, and frequently I see that the modulus of a function has been taken before it is squared.

For example, in a problem I was working on, the following was in the solutions:

$$ \int A(x)A(x)dx = \int|A(x)|^2$$

Why is the modulus sign introduced here? Is it just to get rid of complex values?

I have seen this appear elsewhere as well:

$$\int |f(x)|^2dx < \infty $$

Where the set of all functions $f(x)$ that satisfy the above inequality define a Hilbert Space, why is it necessary to take the modulus before squaring the integrand in order to define a Hilbert Space.

Answer 1

I would say there's something a little more fundamental happening than taking the modulus to deal with complex numbers. Ultimately, squares of sizes/magnitudes are the quantities of physical interest, and that's what the modulus measures. More specifically, for a complex number $z$, $|z|$ is the magnitude of the number and (if $z\neq 0$) $z/|z|$ is the phase. I believe I'm remembering correctly that the phase, in and of itself, has little or no physical significance. Phase differences matter, but the phase itself does not, in the sense that if we replace a wave function $\phi$ with $\epsilon \phi$, where $|\epsilon|=1$, it's still a solution to the same linear equation(s).

So the reason we take the modulus before squaring is because the squared modulus (or the integral thereof) is one of the quantities of intrinsic physical interest.

But if we ask why isn't it equivalent to drop the modulus, then yes, the answer is because we're dealing with complex numbers. And there are numerous reasons why quantum mechanics cannot be done if we limit ourselves to real numbers. Complex numbers naturally appear in Schrodinger's equation. Eigenvalues/eigenvectors are fundamental in quantum mechanics, and you can't get a good spectral theory without going into the complex numbers. Position/momentum are related through the Fourier transform, which also necessarily moves into the complex numbers.