Operations on a State in Different Bases

[danmay]

Say we have the same state \boldsymbol{\psi_r} in momentum basis, or \boldsymbol{\phi_p} in position basis. I want to make either a position observation or a momentum observation. How do I write the operation and the result mathematically, \mathbf{r} \boldsymbol{\psi}, \mathbf{r} \boldsymbol{\phi}, \mathbf{p} \boldsymbol{\psi}, or \mathbf{p} \boldsymbol{\phi}? In terms of the results, which ones would be physically equivalent, assuming the same wavefunction always collapses to the same eigenstate of whatever the observable is. Please ask questions / offer suggestions if I'm not making any sense, because I have a feeling some of it may not.

By the way, is the same as ? Is there a thread/post/sticky on how to use these markups?

 

[Matterwave]

tex gives you a new line.
itex makes the latex show up in the same line of text as your writing.

The different operators have different forms in different bases. The position operator says "multiply by x" if you are in the position basis, but becomes i\hbar\partial_p in the momentum basis. Similarly for the momentum operator.

 

[danmay]

The different operators have different forms in different bases. The position operator says "multiply by x" if you are in the position basis, but becomes i\hbar\partial_p in the momentum basis. Similarly for the momentum operator.

So, for any particular operator, a change of basis would give different mathematical forms to its eigenstates (I assume physically they don't change)? But the eigenvalues would still be the same, right, since they have to be the possible outcomes of observation?