Probability of simultaneous measurements of momentum and position

[Karl G.]

Query:
Given a three- dimensional wavefunction (phi) (x, y, z),
what is the probability of simultaneously measuring
momentum and position to obtain the results
a < y < b and p' < Pz < P" ?
I know that integration of the square norm of the wavefunction of the region
under question yields the probability for finding the position or momentum
of the system described by the wavefunction. But how do you do this
for simultaneous measurements of momentum and position?

Thanks!

 

[ZapperZ]

What do you mean by "simultaneous measurement"? One single measurement that immediately produces both values of the observable? Don't you think that if this is the case, then it doesn't matter how we proceed with the sequence of AB versus BA for non-commuting observables?

Zz.

 

[meopemuk]

Query:
Given a three- dimensional wavefunction (phi) (x, y, z),
what is the probability of simultaneously measuring
momentum and position to obtain the results
a < y < b and p' < Pz < P" ?
I know that integration of the square norm of the wavefunction of the region
under question yields the probability for finding the position or momentum
of the system described by the wavefunction. But how do you do this
for simultaneous measurements of momentum and position?

Thanks!

x, y, and pz are three mutually commuting observables, so it would be convenient to write your wave function in the corresponding basis (phi)(x, y, pz). (To do that, you can perform a Fourier transform of your position-space wave function (phi) (x, y, z) on the variable z.) The next step would be to take the square of the modulus |(phi)(x, y, pz)|^2 and integrate it on the given region of y and pz.

 

[Karl G.]

Sorry to ZapperZ for ambguities in phrasing. However, meopemuk's answer was the one I was looking for. Thanks!