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Changing variables in the Schrodinger equation

  1. Apr 9, 2013 #1
    Suppose I have a Schrodinger equation for two interacting particles located at x and y; so, something like
    [tex]
    \left( i \frac{\partial}{\partial t} + \frac{1}{2m_x} \frac{\partial^2}{\partial x^2} + \frac{1}{2m_y} \frac{\partial^2}{\partial y^2} + V(x-y) \right) \psi(x,y,t) = 0.
    [/tex]
    Now, I want to shift to center of mass coordinates R and r and write the Hamiltonian in terms of them. The resulting Schrodinger equation looks something like
    [tex]
    \left( i \frac{\partial}{\partial t} + H(R,r) \right) \varphi(R,r,t) = 0.
    [/tex]
    Question: is it true that
    [tex]
    \psi(x,y,t) = \varphi \left(\frac{m_x x + m_y y}{m_x + m_y},x-y,t\right)?
    [/tex]
    If so, why? I have been trying to prove this but have so far just been going around in circles it seems...
     
  2. jcsd
  3. Apr 10, 2013 #2

    DrClaude

    User Avatar

    Staff: Mentor

    Can you write ##x## and ##y## in terms of ##R## and ##r##, such that you have a one-to-one mapping between the two? If yes, then there is no difference between expressing a function in terms of ##(x,y)## or of ##(R,r)##.
     
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