# What is the von Neumann equation?

## What is the von Neumann equation?

The so-called driven Liouville–von Neumann equation is a dynamical formulation to simulate a voltage bias across a molecular system and to model a time-dependent current in a grand-canonical framework.

## How do you find the density of a matrix?

At infinite temperature, all the wi are equal: the density matrix is just 1/N times the unit matrix, where N is the total number of states available to the system. In fact, the entropy of the system can be expressed in terms of the density matrix: S=−kTr(ˆρlnˆρ).

What do you mean by density matrix?

In quantum mechanics, a density matrix is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using the Born rule.

Why do we need density matrix?

So why would we need the density matrix? It is a practical tool when dealing with mixed states. Pure states are those that are characterized by a single wavefunction.

### Is density matrix symmetric?

Note that prob- lems of the RDM symmetry properties were greatly simplified, since in most cases of calculating the matrix elements of the density matrix only its completely symmetric component. However, it is not always the case. So in general, the eigenfunctions of RDM possess mixed symmetry.

### Who introduced density?

Archimedes
Scientists measure density by dividing the mass of something by its volume (d = m/v). This is a story about how the concept of density was first “discovered.” around 250 B.C. The King of Syracuse, where Archimedes lived, thought that he was being cheated by the metal craftsman who made his golden crown.

Is partial trace linear?

The first and foremost thing to realize is that the partial trace over a density matrix is indeed a linear CPTP map Λ, but it is not a map from any Cn×n→Cn×n (i.e. to `itself’ – the same dimension), but rather to a density operator space with a lower dimension: Cn×n→Cm×m with m

Is the density operator part of the Liouville-von Neumann equation?

Infroduction tation of measurement in quantum mechanical systems . The density operator is the counter- The Liouville—von Neumann equation is the part of the classical distribution function [5—7]. basic framework unifying the quantum mechani- The same operator emerges from the quantum cal and statistical descriptions of matter.

## Is the von Neumann equation immediate to prove?

Well, the von Neumann equation holds witin Schroedinger pictureand it is immediate to prove in quantum physics (differently from Liouville equation which needs to preventively establish the non-trivial Liouville theorem for the symplectic measure on the space of phases).

## Is the assumption of linearity replaced with non-contextuality?

Gleason’s theorem shows that in Hilbert spaces of dimension 3 or larger the assumption of linearity can be replaced with an assumption of non-contextuality. This restriction on the dimension can be removed by assuming non-contextuality for POVMs as well, but this has been criticized as physically unmotivated.