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Confusing variable change calculation

  1. Feb 5, 2012 #1
    Suppose you start with a system of N particles identified by position vectors [itex]r_1, r_2, \ldots, r_N[/itex] and masses [itex]\mu_1, \mu_2, \ldots, \mu_N[/itex]. Then (quantum mechanically) the kinetic energy operator for this system is given by (assuming [itex]\hbar = 1[/itex])

    [tex]
    T = \sum_{i = 1}^N -\frac{\Delta_i}{2\mu_i},
    [/tex]

    where [itex]\Delta_i = \Delta_{r_i}[/itex] is the Laplacian in the variable [itex]r_i[/itex]. Now, let's make the change of variables [itex]\eta_i = r_i - r_N[/itex]. According to the book I have, the kinetic energy in the new variables looks like

    [tex]
    T' = \sum_{i = 1}^{N-1} -\frac{\Delta_i}{2\mu_i'} + \sum_{i < j} \nabla_i \cdot \nabla_j,
    [/tex]

    where [itex]\Delta_i = \Delta_{\eta_i}[/itex], [itex]\nabla_i = \nabla_{\eta_i}[/itex], and

    [tex]
    \frac{1}{\mu'_i} = \frac{1}{\mu_i} + \frac{1}{\mu_N}.
    [/tex]

    I simply do not see how this is true. It seems that, if one applied the chain rule, one obtains [itex]\Delta_{\eta_i} = \Delta_{r_i}[/itex]; is that not true?
     
  2. jcsd
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