Graduate Worked Out Example of Consecutive Observations in QM

[boderam]

I know that in QM, one observation like position will alter the wavefunction so that momentum changes. But how do we see this mathematically when we include time dependence, whether in matrix mechanics or wave mechanics? Is it as simple as writing PQx where x is the state, Q position matrix, P momentum matrix? How do we include the time dependence in the matrices, i.e. so the P measurement is done t seconds after Q? Thanks.

 

[arkajad]

There are several methods. They differ by the way "observation" is described. Some of them are consistent, other not so. Notice that Qx has very little to do with "observation of the position". You need to have a model of a measurement first. Then you need to find out whether your model is compatible with what is being observed in reality.

An approximate effect of the action of a position detector put at the point x=a and activated at the moment t=t0, and only at that moment, will be changing the Schrödinger wave function at t=t0 by multiplying it by a Gaussian function of x centered at x=a, and of the width corresponding to the resolution of the detector. After that you can continue your Schrödinger evolution. But that is only an approximation, not a very realistic one, if only because no real detector is activated just for an instant of time of zero width.

 

[boderam]

There are several methods. They differ by the way "observation" is described. Some of them are consistent, other not so. Notice that Qx has very little to do with "observation of the position". You need to have a model of a measurement first. Then you need to find out whether your model is compatible with what is being observed in reality.

An approximate effect of the action of a position detector put at the point x=a and activated at the moment t=t0, and only at that moment, will be changing the Schrödinger wave function at t=t0 by multiplying it by a Gaussian function of x centered at x=a, and of the width corresponding to the resolution of the detector. After that you can continue your Schrödinger evolution. But that is only an approximation, not a very realistic one, if only because no real detector is activated just for an instant of time of zero width.

This all is very vague, I find a lot of explanation lacking. I am looking for some sort of formulation where we can have an observation operator of some sort, so when we look at it together with the wave function we have the evolution of the wave function by being changed by the observation. It seems to me that this sort of idea would be well developed yet I haven't seen it very much. Perhaps it is that most of QM is applied to single observations and has no need for such a theory as I am describing? Could you point me to a reference on what you were describing?

 

[arkajad]

It would be unfair for me to suggest checking my own papers. Just gave you a vague alas neutral reply, hoping that someone else will give you a more objective answer.