What is the connection between energy eigenstates and position?

[jeebs]

The first thing I remember hearing about in QM was the time-independent 1-D Schrödinger equation, $Hψ = (\frac{-\hbar^2}{2m}\frac{d^2}{dx^2} + V)ψ(x) = Eψ(x)$. This is an eigenvalue equation, the Hamiltonian operator H operating on the energy eigenstate ψ to produce the product of the energy eigenvalue, E, and ψ.

However, we also come to know this state ψ by another name, the "wavefunction", and we find that if we take |ψ(x)|^2 we find the probability of finding our particle at at position x.

My question is, what is it about the eigenstates of the energy operator in particular that should mean we can find out this information about the likelihood of a particle occupying a certain position x upon measurement? I don't see the connection - especially seeing as we could take a free particle (V=0) so that the energy of the particle has no dependence on position, only momentum?
In other words, why don't we take any other eigenstate for any other observable quantity, square that and use that for our position probability?

 

[Matterwave]

You can choose to express your wavefunction in any complete basis. The thing that's special about the energy eigenbasis is it gives you an easy way to evolve the states in time. The Born postulate has nothing to do with energy eigenbases.

 

[sleventh]

the wave function solutions of the Schrödinger equation for any system are solutions in a "state space" within the Hilbert Space. The Hilbert space is a space where the elements of the space are solutions to the wave equation (where the operation is just the inner product). A state of a quantum mechanical system is then a vector in the Hilbert space, and observables (which act as operators in Quantum mechanics) are a type of linear operator. Like any linear operator there exists a matrix representation allowing for our eigen value to be relevant. But the fact that these eigen-values correspond to the systems energy comes solely from the derivation of the Schrödinger equation which uses De Broglie's relations and the least action principle to find a wave form from particle quantisation. The fact energy became the scalar acting as a eigen-value for an eigen-value equation was a beautiful by-product.
The state space is not limited to a position representation. ANY observable my act as the linear operator in our Hilbert space.