- #1

- 325

- 3

**Why do we associate the energy eigenstates with the "wavefunction" and "position"?**

This is something that has bothered me for a long time but that I never got around to asking. I suspect I'll feel like an idiot for not knowing this but anyway...

One of the first things I was introduced to in quantum mechanics courses is the time-independent "schrodinger" equation, or in other words, the "energy eigenvalue equation", [tex] H\Psi = E\Psi [/tex], where [tex]\Psi[/tex] are the eigenfunctions of H.

We were also introduced to the idea that you could take this object [tex] \Psi [/tex], square it, and that would give you the probability of finding your particle at a given position. [tex]\Psi[/tex] gets called the "wavefunction" but I notice that the use of that word seems to dry up when you start getting into doing your quantum mechanical calculations using Dirac notation.

I have never understood where this dual use of the energy eigenstate [tex]\Psi[/tex] came from, and it seems like it should be an important thing to know. I mean, we have our set of observables like momentum, energy, etc. and their corresponding operators, eigenstates and eigenvalues - why is it the energy eigenstates in particular that we take as our "wavefunction"?

We could find different eigenstates for any operator that does not commute with H, right?

Why isn't our "wavefunction" the eigenstate of some other operator?

This can't have been an arbitrary choice?