Mixed State Treatment: Eigenvalue Degeneracy

[jk22]

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 : $$\sum_{\lambda}\int \rho (\lambda,\alpha)|\lambda, \alpha\rangle\langle \lambda, \alpha|d\alpha $$ ?

 

[A. Neumaier]

$P_E\rho P_E$, where $P_E$ is the orthogonal projector to the eigenspace of energy $E$.

 

[jk22]

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 ?

 

[A. Neumaier]

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.

In fact there never exist pure state at non zero temperature ?

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.