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Time evolution of ensembles

  1. Dec 31, 2003 #1

    hellfire

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    The density operador which describes an ensemble, [tex]
    \rho = \sum_{i} w_i |a_i> <a_i|[/tex] (represented in the basis in which it is diagonal), evolves in time such that [tex]|a_i(t)> = U(t) |a_i> [/tex]when the enemble remains undisturbed, i.e. the w_i do not change.

    But for irreversible processes the w_i may change towards a more random ensemble, driving the density matrix to a diagonal form with equal values for the w_i.

    This is what I got after reading chapter 3.4 of Sakurai´s Modern Quantum Mechanics. But what Sakurai does not explain is how the time evolution of the w_i may look like for irreversible processes. May be someone can give a hint or a reference.

    The background of my question: I´ve read that a transition from a pure ensemble (density matrix has only one element different from zero) to a mixed ensemble (with several w_i different from zero) is not allowed in QM (I read this in relation with black holes). I would like to understand why.

    Thanks.
     
  2. jcsd
  3. Jan 2, 2004 #2

    Another God

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    Sorry that I can't help you with your question at all, but is it at all possible that you could help me out a little, and explain your question a little?

    SImply explain the variable in the equations, explain what the equation is for, what each part of it represents etc... Maybe if you help me understand the equations, I may be able to eventually help you?

    (OK, so maybe thats a little unlikely...but I'll try.)
     
  4. Jan 2, 2004 #3

    hellfire

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    I am not sure whether I am able to be more clear with this question, but I will try (although I am afraid I will still repeat).

    As you may know, an ensemble can be described with a density operator [tex]\rho[/tex] such that [tex]\rho = \sum_{i} w_i |a_i> <a_i|[/tex] (represented in the basis in which it is diagonal), where each of the w_i is a real number representing the relative population of elements in a given coherent state [tex]|a_i>[/tex] and such that [tex]\sum_{i}w_i = 1[/tex].

    In ensembles which remain undisturbed (relative populations remain constant) the [tex]|a_i>[/tex] evolve in time affected by the time-evolution operator [tex]|a_i(t)> = U(t) |a_i>[/tex] (Schroedinger picture).

    This is what I read in Sakurai´s book. Now my question.

    I assume (but I am not really sure) that in ensembles which do not remain undisturbed (physical processes which are not reversible), the time evolution may be described as a change of the w_i. The entropy is defined as [tex] S = - k \sum_{i} w_i ln w_i [/tex]. How does entropy increase otherwise, if the w_i do not evolve?.

    Now, if the w_i may evolve, why is a transition between a pure ensemble (only one w_i) and a mixed ensemble (several w_i) not possible?
     
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