The Probability Law for Generic Density Operators

E. L. Rumyantsev, P. E. Kunavin

We present systematic construction of probability and probability current densities operators for one-band single particle Pauli equations starting from the operators in Dirac electron model within Second Quantized Approach. These operators are of importance as in probability interpretation of experimental data, so in establishing of boundary conditions. It is shown that derived operators differ significally from their convential Schrodinger-type counterparts. The generalization of continuity equation for probability density under external perturbations and physical meaning of additional source terms is discussed. The presented approach can be useful in analysis of carriers dynamics described within generic multicomponent $\vec{k} \cdot \vec{p}$ Hamiltonians in Envelope Function Approximation (EFA).

Ravinder Rupchand Puri

Fedor Herbut

Zurek's derivation of the Born rule from envariance (environment-assisted invariance) is retraced in an attempt to present a more detailed derivation, which is extended to encompass the trace-rule form of the quantum probability law in its full generality. The investigation begins by a review presentation of Schmidt decomposition, and a detailed and complete theory of twin unitaries, which are the other face of envariance, and which stand in close connection with twin observables. The trace rule tr( E ρ), with E being an event (projector) and ρ a quantum state (density operator), is derived in five stages. The first three achieve the same as Zurek's derivation of Born's rule, but in a different way, using the full power of twin unitaries. Stage 4 extends the Zurek-like derivation to states (ray projectors) that are not eigenstates of the density operator. Finally, stage 5 utilizes the assumption of continuity of the probability law in the density operators to extend the trace rule to isolated (not correlated) systems.

Mingzhe Li, Lixuan Chen, Chuanhong Chen

An ideal gas trapped in a generic power-law potential in any dimensional space is studied. The density of states of this system is derived. Both Fermi and Bose gases are discussed, and their properties are derived straightforwardly from the density of states.

Philip L. Bowers