How do i perform a measurement of a many body system?

[spocchio]

first of all, I'm a new bie of QM,

with q1,p1 and q2,p2 the couples of position and momentum
THE QUESTION IS: can I just measure a $$\hat{q1}$$ with no measure of $$\hat{q2}$$? and how? and how I calculate the new wave function with the dependece of q2?

I hope that $$\hat{q1}|q1,q2>=q1|q1,q2>$$ is true
if so when i have a certain state $$|\phi>$$
and I measure q1 i'll obtain a new state like $$|phi>=|q1,q2>$$ with a probability of
$$|<q1,q2|\phi>|^2$$
but wath q2 have I to choose?
but, does it have sense? the q2 dependece vanish!? where I'm wrong?

it could seems a bit confusing, sure as i am.

 

[nnnm4]

You're touching on what is called symmetrization of the wave function. The probability distribution associated with the many-body state is given by |u(x1,x2)|^2 whe u is the many body wave function. We require that this be the same upon exchange of particle label, because we cannot in principle label quantum mechanical particles. That means |u(x1, x2)|^2 = |u(x2,x1)|^2. So it doesn't matter which position operator you use, they're all equivalent. The condition of the wave function itself is not unique mathematically. I.E. we could have u(x1,x2) differ from u(x2,x1) by some complex phase. However, in 3-dimensions symmetries of space-time require that u(x1,x2)=u(x2,x1) or u(x1,x2) = -u(x2,x1) (Ihave neglected the spin symmetries associated with the state. The former is true for bosons and the latter for fermions.

 

[spocchio]

Oh, you are just making things more complicated...
as I know these problems of symmetries are forced in QM, they don't came from any
principal postulate.
I don't want to deal about these symmetries problems.
I think this problem of many bodies is mathematically equivalent to a problem of 1 body in N dimensions..
for example q1=x,q2=y for a free particle in 2D

so, the new version of the problem is:
for a particle in N Dimension, how do I perform a measurement in only one coordinate?
for example, with a slit.

 

[nnnm4]

That's incorrect. For a general system the wavefunction is a nonseperable scalar function of 3N arguments and contains all of the relevant information about the state of the system given the Hilbert space.