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 DensityMatrixToStateMultipole

Source Notebook

Represent matrix elements via density state multipoles (statistical tensors)

Contributed by: Xu-xing Geng and Gao-xiang Li
(Department of Physics, Central China Normal University)

ResourceFunction["DensityMatrixToStateMultipole"][j1,j2,m1,m2]

expands the m1,m2 density matrix element corresponding to angular momenta j1 and j2 into state multipoles.

Details

DensityMatrixToStateMultipole expands the density matrix element corresponding to the angular momentum J1=J2=J to state multipoles corresponding to the q-components of an SO[3] representation tensors of rank k.
The state multipole is defined as the trace of the SO[3] representation tensors: , where T is the SO[3] representation tensors and tois the density matrix.
The expansion of the density matrix of angular momentum into the state multipole corresponding to the SO[3] representation tensors has important applications in the field of quantum mechanics. State multipole expansion represents the density matrix as a linear combination of a series of SO[3] representation tensors operators. This method is very useful in dealing with problems related to angular momentum, especially in describing the symmetry of the system, calculating the expectation value, and analyzing the coherence and entanglement of quantum states.

Examples

Basic Examples (1) 

Expand the density matrix element of the angular momenta J1=1 and J2=2 into state multipoles:

In[1]:=
ResourceFunction["DensityMatrixToStateMultipole"][1, 2, 1, 0]
Out[1]=

Scope (4) 

Expand the density matrix element into state multipoles:

In[2]:=
ResourceFunction["DensityMatrixToStateMultipole"][5, 4, 1, 1]
Out[2]=

Expand all density matrix elements into state multipoles:

In[3]:=
Table[Subscript[\[FormalRho][1, 1], m, n] == ResourceFunction["DensityMatrixToStateMultipole"][1, 1, m, n], {m, -1, 1}, {n, -1, 1}] // TableForm
Out[3]=

Expand all density matrix elements :

In[4]:=
Table[Subscript[\[FormalRho][1/2, 1/2], m, n] == ResourceFunction["DensityMatrixToStateMultipole"][1/2, 1/2, m, n], {m, -1/2, 1/2}, {n, -1/2, 1/2}] // TableForm
Out[4]=

Expand all density matrix elements :

In[5]:=
Table[Subscript[\[FormalRho][1/2, 3/2], m, n] == ResourceFunction["DensityMatrixToStateMultipole"][1/2, 3/2, m, n], {m, -3/2, 3/2}, {n, -3/2, 3/2}] // TableForm
Out[5]=

Publisher

Xuxing geng

Requirements

Wolfram Language 13.0 (December 2021) or above

Version History

  • 1.0.0 – 27 September 2024

Source Metadata

  • Citation:
    Blum K. Density matrix theory and applications[M]. Springer Science & Business Media, 2012.

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