Structure of the density matrix providing the minimum generalized uncertainty relation for mixed states
M U Karelin, A M Lazaruk, M U Karelin · Journal of Physics A Mathematical and General · 2000
For a configurational space of arbitrary dimension a strict form of the uncertainty principle has been obtained, which takes into account the dependence of the inequality limit on the effective number of pure states present in a given statistical mixture. It is shown that in a state with minimal uncertainty the density operator eigenfunctions coincide with the stationary wavefunctions of a multi-dimensional harmonic oscillator. The spectrum of eigenvalues of the density matrix is degenerate and its terms with the same uncertainty have identical weights in the expansion. This leads to additional `packing of states' (i.e. decreasing of the specific phase volume per particular state in the mixture as compared with the limit of the system in a pure state). The analogy between the generalized uncertainty principle and one of the main postulates of statistical physics on phase-space partitioning into cells corresponding to one quantum state is discussed.