The explanation depends on how deep you want to go into the mathematics of it. The simplest explanation is that the uncertainty principle means you can't get a single number for p if you have a single number for x. So if your operator for position is just "x," your operator for momentum has to be messier. I think the better explanation is to dig just a little deeper and understand that your wavefunction could either be as a function of position or as a function of momentum: you can write it as either Ψ(x) or Ψ(p), and the form your operators take depend on the choice of this variable.

If you define your wavefunctions as functions of position, then the operators you use are:

**x**=x

**p**=-i*(h/2π)*∂/∂x

**Ψ**=Ψ(x)

Boldface here is being used for the general operators and wavefunction, and non-boldface is being used for operators once you choose to work as functions of position or functions of momentum.

But you can also think of them as functions of momentum, in which case:

**x**=i*(h/2π)*∂/∂p

**p**=p

**Ψ**=Ψ(p)

It turns out these are the relations that satisfy the uncertainty relation. If you're more mathematically savvy, they're all tied up in Fourier transforms, which is how you change from variables x to p.