# Mixed state treatment

1. Nov 18, 2015

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

2. Nov 18, 2015

### A. Neumaier

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

3. Nov 18, 2015

### 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 ?

Last edited: Nov 18, 2015
4. Nov 19, 2015

### 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.

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.