Main Question or Discussion Point
Why do we have to use operators in QM?
What do you mean by "a mathematical formalism" ?B/c it works. It is a mathematical formalism which successfully describes nature. I don't think that we can explain this based on a deeper reason.
How are Laplace and Fourier transforms operators?We are also using operators in classical mechanics. For instance when solving differential equations we use sometimes Laplace or Fourier transform. Laplace and Fourier transforms are operators. Every function on the phase space is an operator (multiplication operator) - they commute. Translations and rotations acting on such functions are operators - they do not commute.
Yes I have met Fourier transforms.Fourier transform F applied to a function f gives you another function F(f). The map is linear. So, you gave a linear operator. By the Plancherel's theorem this is a unitary operator. Moreover its square is the inversion:
(F^2 f)(x) = f(-x)
Very interesting operator. But I am not sure if being a second year physicist you have already met with Fourier's transform?
Would you please elaborate?All of these quantities can be collectively grouped together and interconnected in quantum they are grouped together by the wave function for a given system. These quantities can be seen by using certain operators on the wave function.
I don't really get the last two sentences.It's not just in QM that we use operators. The only thing that changes is that you have to take extra caution with the operators you do use in QM. That's really the motivation for learning a little bit of abstract algebra in QM.