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Conservation of 4-Momentum

  1. Feb 23, 2017 #1
    This problem assumes working in natural units where ##c=1##, and using the Minkowski metric where the time component is positive and the space ones negative (as I know the opposite convention is just as commonly used).

    EDIT: I had intended to display 4-vectors as bold and the 3-vectors with the arrow but forgot that it wont parse the Tex properly. So I left the 3-vectors with the arrows, and if it has no arrow and has no component subscript (i.e. x/y/z) assume its a 4-vector.

    1. The problem statement, all variables and given/known data

    Particle u of mass ##m_u## has total energy ##E_u## in the the frame of the lab. It decays into two new particles, particle v of mass ##m_v## and particle w of mass ##m_w##.

    Show that in particle u's rest frame, the energy of outgoing particle v is given by ##E_v = \frac{m_u^2+m_v^2-m_w^2}{2m_u}##


    2. Relevant equations
    ##p \cdot p = m^2##
    ##E^2 = m^2 + |\vec{p}|^2##

    3. The attempt at a solution
    Never done conservation of four momentum problem before, but I assume that it is the case that ##p_u = p_v + p_w## and with the energy components being conserved, and the 3-momentum components being conserved, so that ##p_u = (E_v + E_w, \vec{p_v} + \vec{p_w})## ?

    So far I have done,

    [tex]
    p_w = p_u - p_v \\
    p_w \cdot p_w = (p_u - p_v) \cdot (p_u - p_v) \\
    m_w^2 = m_u^2 + m_v^2 - 2 ( p_u \cdot p_v ) \\
    p_u \cdot p_v = \frac{m_u^2+m_v^2-m_w^2}{2}
    [/tex]

    Which implies that ##p_u \cdot p_v = E_v m_u ## but this is where I come unstuck and cant get that result.

    I have that ##p_u = (E_v + E_w, \vec{p_v} + \vec{p_w}) = (E_u , \vec{p_u}) ##
    and that ##p_v = (E_v , \vec{p_v}) ##

    Doing the dot product I get,
    [tex]
    p_u \cdot p_v = E_v(Ev + E_w) - \vec{p_v}(\vec{p_v}+\vec{p_w}) \\
    p_u \cdot p_v = E_v^2+E_v E_w - p_{v_x}^2 - p_{w_x}^2 - p_{v_y}^2 - p_{w_y}^2 - p_{v_z}^2 - p_{w_z}^2
    [/tex]

    I am not sure if this is the correct way to go about it, but I cant seem to progress from here, not sure if it is because I have gone down the wrong path or if its just a algebra/connection issue. Any help/advice is much appreciated, thanks :)
     
    Last edited: Feb 23, 2017
  2. jcsd
  3. Feb 23, 2017 #2

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Nice

    This follows easily if you keep in mind that you are in the rest frame of particle u.
     
  4. Feb 24, 2017 #3
    Thanks for reminding about that, managed to get it now, since ##p_u = (E_u, 0)=(m_u,0)##, Cheers :)
     
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