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Q:Is it possible to do a coordinate transfomation in momentum space?

  1. Sep 9, 2014 #1
    Q: How does one do a coordinate transformation in momentum space while insuring conservation of momentum?

    I have a several particles with momentum components [itex] P_x , P_y , P_z [/itex].
    I would like to rotate the x, y, and z axis. By angle θ in the x/y and angle Θ in the y/z .
    So giving new momentum [itex] P_x' , P_y' , P_z' [/itex].

    Is it possible to do this an conserve momentum while remaining in the lab frame? (The particles are relativistic but I don't believe this matters). What are the coordinate transformations?
  2. jcsd
  3. Sep 9, 2014 #2
    The conserved quantity is a vector, you can rotate the basis vectors however you want and it wouldn't change the magnitude of the vector.
  4. Sep 9, 2014 #3
    just to double check [itex] P^2=P_x^2 +P_y^2 +P_z^2=P_x'^2 +P_y'^2 +P_z'^2 [/itex] correct?
    Last edited by a moderator: Sep 9, 2014
  5. Sep 9, 2014 #4


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    Staff: Mentor

    If you've done the transformation to the primed coordinates correctly, yes.

    In fact, HomogeneousCow has understated how much is conserved; the direction of the vector is also conserved. Of course it's a bit tricky talking about the "direction" of a vector when you don't have coordinate axes to make angles with - (1,0) in coordinates in which the x-axis points to the northeast is the same vector in the same direction as ##\sqrt{2}/2(1,1)## in coordinates in which the x-axis points east, but it's not obvious at all from the coordinates that that is so.

    However, the dot-product of two vectors is invariant under these coordinate transformations, and as the dot-product depends on the angle between the vectors, that gives us a coordinate-independent way of claiiming that direction is also invariant.
    Last edited: Sep 9, 2014
  6. Sep 9, 2014 #5
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