Stamp density

The matrix density , or operator density is a mathematical entity introduced by the mathematician and physicist John von Neumann. It makes it possible to summarize in only one matrix all the possible unit of the quantum states of a physical system given to a given moment, thus marrying quantum Mécanique and Physique statistics.

Definition

Pure case

The description of the system is done here thanks to a vector of state $|\; \backslash \; psi\; (T)\; \backslash \; rangle$ which one can develop on the bases $\backslash \; \{\; |\; u\_n\; \backslash rangle\; \backslash \}$ :

$\backslash \; left|\; \backslash \; psi\; (T)\; \backslash \; right\; \backslash \; rangle\; =\; \backslash \; sum\_n\; \{c\_n\; (T)\; \backslash \; cdot\; \backslash \; left|\; u\_n\; \backslash right\backslash rangle\}$

with $\backslash \; sum\_n\; |\; c\_n\; (T)\; |^2\; =\; 1\; \backslash ,$

The operator density is defined for a pure state by:

$\backslash \; hat\; \backslash \; rho\; =\; |\backslash \; psi\; (T)\; \backslash \; rangle\; \backslash \; langle\; \backslash \; psi\; (T)\; |\; =\; \backslash \; sum\_\; \{N,\; p\}\; c\_n^*\; (T)\; c\_p\; (T)\; |\; p\; \backslash \; rangle\; \backslash \; langle\; N\; |$

Mix statistical pure states

By admitting that a certain physical system can be, at a certain moment T, in a statistical mixture (finished or infinite) of quantum states $|\; \backslash \; psi\_i\; \backslash \; rangle$ with probabilities $p\_i$ (where $\backslash \; sum\_i\; p\_i\; =\; 1\; \backslash ,$), then the matrix density representing the unit of these states is:

$\backslash \; hat\; \backslash \; rho\; =\; \backslash \; sum\_i\; p\_i\; |\backslash psi\_i\; \backslash rangle\; \backslash langle\; \backslash psi\_i\; |$

The statistical aspect introduces here is of two nature, one traditional and the other quantum one:

1. traditional : had with the estimate of the ket by a statistical distribution of different the kets possible,

2. quantum : quantum uncertainty fundamental even if the system is perfectly given.

The elements of the matrix density are worth:

$\backslash \; hat\; \backslash \; rho\_\; \{pn\}\; =\; \backslash \; sum\_i\; p\_i\; \backslash \; langle\; u\_p^\; \{(I)\}\; |\; \backslash hat\; \backslash rho\_i\; |\; u\_n^\; \{(I)\}\; \backslash \; rangle\; =\; \backslash \; sum\_i\; p\_i\; c\_n^\; \{(I)\; *\}\; c\_p^\; \{(I)\}$

Property

The matrix obtained has the following properties:

• It is square, $\backslash \; hat\; \backslash \; rho\; =\; \backslash \; hat\; \backslash \; rho^\; \{\backslash \; dagger\}$, it can thus be diagonalized, and its eigenvalues are positive.
• Its trace is equal to 1, $Tr\; (\backslash \; hat\; A)\; =1$, conservation of the total probability.
• It must be definite positive or null.
• In the case of a pure state, the operator density is then a Projecteur: $\backslash \; hat\; \backslash \; rho^2\; =\; \backslash \; hat\; \backslash \; rho$.
• $Tr\; (\backslash \; rho^2)\; \backslash \; the\; 1$, with equality if and only if the physical system is in a pure state (i.e. all the $p\_i$ are null except one).

Median value

One can calculate the median value of a Observable has starting from the formula:

$\backslash \; langle\; \backslash \; hat\; has\; \backslash \; rangle\; =\; \backslash \; langle\; \backslash \; Psi\; |\backslash \; hat\; has\; |\; \backslash \; Psi\; \backslash \; rangle\; =\; Tr\; (\backslash \; hat\; has\; \backslash \; hat\; \backslash \; rho)\; =\; Tr\; (\backslash \; hat\; \backslash \; rho\; \backslash \; hat\; A)$

with $\backslash \; hat\; \backslash \; rho\; =\; \backslash \; sum\_i^N\; p\_i\; \backslash \; hat\; \backslash \; rho\_i$ is the matrix density of a statistical mixture of states.

Evolution with time

See also: Operator of evolution

The temporal evolution of the operator of state is given by the equation of Schrödinger dependant on time:

$\backslash \; hat\; H\; \backslash \; left|\; \backslash \; Psi\; (T)\; \backslash \; right\; \backslash \; rangle\; =\; I\; \backslash \; hbar\; \{D\; \backslash \; over\; dt\}\; \backslash \; left|\; \backslash \; Psi\; (T)\; \backslash \; right\; \backslash \; rangle$

Bond with the entropy

Lastly, one can define the Entropie Von Neumann:

$S=-k\_B\; Tr\; (\backslash \; hat\; \backslash \; rho\; \backslash \; ln\; (\backslash \; hat\; \backslash \; rho))$

where $k\_B$ is the Boltzmann constant.

The entropy of a pure state is null, because there is no uncertainty on the state of the system. One can also find a base where the matrix is diagonal, with 0, and one 1 on the diagonal, which gives an entropy well equal to 0.

See too

• quantum State

• Physical quantum statistics
• Mechanical

References

Category: Quantum physics

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