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The definition of the density operator in Pathria

  1. Sep 30, 2012 #1
    Hello Everybody,

    I am working through Pathria's statistical mechanics book; on page 114 I found the following definition for the density operator:
    [tex] \rho_{mn}= \frac{1}{N} \sum_{k=1}^{N}\left \{ a(t)^{k}_m a(t)^{k*}_n \right \}, [/tex]
    where N is the number of systems in the ensemble and the a(t)'s are expansion coefficents.
    Now my question is: what does this definition mean? Especially the term [tex] a(t)^{k}_m a(t)^{k*}_n. [/tex] I do not get it.

    Any help would be greatly appreciated!
     
  2. jcsd
  3. Oct 8, 2012 #2

    Jano L.

    User Avatar
    Gold Member

    The formula you wrote above refers to the definition of density matrix for a finite ensemble of isolated systems (beware, not for a subsystem in interaction with environment). It works as follows.


    Imagine that there is an ensemble of N copies of the system. Denote the wave function describing the k-th system by

    [tex]
    \psi^{k}(\mathbf r).
    [/tex]

    (Different copies have different wave functions).


    It is assumed that this function can be expressed as a discrete linear combination of some basis functions [itex]\Phi_m[/itex], which are the same for all k:


    [tex]
    \psi^{k}(\mathbf r) = \sum_k a_m^k \Phi_m(\mathbf r).
    [/tex]

    The numbers [itex]a^k[/itex] are complex expansion coefficients.

    (Such expansion is possible if the set of functions [itex]\Phi_m[/itex] is complete, like for Hamiltonian eigenfunctions of harmonic oscillator. In case of hydrogen eigenfunctions, things are more complicated, due to continuous spectrum of Hamiltonian).

    The density matrix is introduced usually as a quantity [itex]\rho_{mn}[/itex] that appears in the calculation of average value of some quantity [itex]f[/itex], say energy, over the ensemble.

    The average over the ensemble is the weighted sum

    [tex]

    \langle \langle f \rangle \rangle = \sum_k p_k \langle f \rangle^k,

    [/tex]

    where [itex]p_k = 1/N[/itex] is the probability that the system is in state described by k-th wave function.

    The expression

    [tex]

    \langle f \rangle^k
    [/tex]

    used above is the average of [itex]f[/itex] in a state described by [itex]\psi^k[/itex] function and can be expressed as

    [tex]
    \langle \psi^k | \hat f | \psi^k \rangle = \sum_{m,n} a_m^{k} a_n^{k*} f_{nm}.
    [/tex]

    where [itex]f_{nm} = \langle \Phi_n|\hat f|\Phi_m\rangle[/itex].

    Then, the average over the ensemble is

    [tex]

    \langle \langle f \rangle\rangle = \sum_k \frac{1}{N} \sum_{m,n} a_m^{k} a_n^{k*} f_{nm}.
    [/tex]


    This can be rewritten as

    [tex]

    \sum_m \left( \rho_{mn}f_{nm} \right)

    [/tex]

    where the quantity

    [tex]
    \rho_{mn} = \sum_k \frac{1}{N} a_m^{k} a_n^{k*}
    [/tex]

    was named the density matrix.
     
  4. Oct 8, 2012 #3
    Perfect.

    Many many thanks!
    If you were living in Hamburg, Germany, I would give you a bag full of cookies!
    I am really thankful! :)
     
  5. Oct 8, 2012 #4

    Jano L.

    User Avatar
    Gold Member

    Glad to be of help.
     
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