One of the main reasons why physicists are obsessed by this topic is that position and momentum are mapped to the phase space coordinates of Hamiltonian mechanics, which appear in the matrix mechanics formulation of quantum mechanics, which offers insight into the theory, particularly when generalizing to relativistic quantum mechanics.
One way to think is: what is the definition of space space? It is a way to write the wave function such that:
  • the position operator is the multiplication by
  • the momentum operator is the derivative by
And then, what is the definition of momentum space? It is of course a way to write the wave function such that:
  • the momentum operator is the multiplication by gives the best idea intuitive idea: the Fourier transform writes a function as a (continuous) sum of plane waves, and each plane wave has a fixed momentum.
A way to write the wavefunction such that the position operator is: i.e., a function that takes the wavefunction as input, and outputs another function:
If you believe that mathematicians took care of continuous spectrum for us and that everything just works, the most concrete and direct thing that this representation tells us is that:
the probability of finding a particle between and at time
This operator case is surprisingly not necessarily mathematically trivial to describe formally because you often end up getting into the Dirac delta functions/continuous spectrum: as mentioned at: mathematical formulation of quantum mechanics
One dimension in position representation:
In three dimensions In position representation, we define it by using the gradient, and so we see that
Video 1. Position and Momentum from Wavefunctions by Faculty of Khan (2018) Source. Proves in detail why the momentum operator is . The starting point is determining the time derivative of the expectation value of the position operator.