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Changing the Hamiltonian to a new frame of reference

  1. Mar 19, 2013 #1
    Suppose I'm considering particles of mass [itex]\mu_i[/itex], [itex]1 \leq i \leq 3[/itex], located at positions [itex]r_i[/itex]. Suppose I ignore the potential between [itex]\mu_1[/itex] and [itex]\mu_2[/itex]. Then the Hamiltonian I'd write down would be

    H = -\frac{1}{2\mu_1}\Delta_1 -\frac{1}{2\mu_2}\Delta_2 - \frac{1}{2\mu_3}\Delta_3 + V_1(r_3 - r_1) + V_2(r_3 - r_2).

    But what if I instead want to work in a frame of reference in which [itex]\mu_1[/itex] is at rest? How should I go about changing [itex]H[/itex]? I'm never very sure of myself when I do these kinds of calculations, so any help would be appreciated...thanks!
  2. jcsd
  3. Mar 20, 2013 #2


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    The are several options, but first of all no particle will be "at rest" at the level of the Hamiltonian; it's a special solution (in classical mechanics) where one momentum p vanishes, i.e. where one particle is at rest. You must not set p=0 in H.

    Coordinate changes can be done at three levels
    - on the level of the Lagrangian, as usual in classical mechanics
    - on the level of the Hamiltonian function, i.e. using canonical transformations
    - on the level of the Hamiltonian operator, i.e. using unitary transformations

    There is a nice example using unitary transformations showing what really happens when we introduce the c.o.m. frame for a central potential; the new coordinates can be interpreted as r,R,p,P, where R does not appear in H, so P is conserved. That means tha strictly speaking we should not set P=0, but we have a plane wave in R and P.
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