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Mixed state treatment

  1. Nov 18, 2015 #1
    Suppose i have an eigenvalue which is two fold degenerate-. Is it possible to have a density matrix formulation for the following : there is a continuum of states considered namely every state in the eigenspace.

    How would it be written : $$\sum_{\lambda}\int \rho (\lambda,\alpha)|\lambda, \alpha\rangle\langle \lambda, \alpha|d\alpha $$ ?
     
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  3. Nov 18, 2015 #2

    A. Neumaier

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    ##P_E\rho P_E##, where ##P_E## is the orthogonal projector to the eigenspace of energy ##E##.
     
  4. Nov 18, 2015 #3
    Thanks. What I would like to know is if it is possible to treat every vector in the eigenspace separately ?

    My other question is : since the temperature is never 0K the endstate in a measurement of a multiple eigenvalue is a mixed state. In fact there never exist pure state at non zero temperature ?
     
    Last edited: Nov 18, 2015
  5. Nov 19, 2015 #4

    A. Neumaier

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    States with definite energy usually are intrisically mixed states unless there are other conserved quantities that allow one to split the eigenspace into smaller invariant subspaces.

    Yes. By definition, a state at fixed positive temperature has a density matrixof the form ##e^{-S/\kbar}## with an operator $S$ whose expectation is the entropy.

    Most states in nature are mixed; only systems with very few degrees of freedom can be prepared in a pure state.
     
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