“9-j symbol”的意思、由来-开放百科全书

词条

9-j symbol

释义

Recoupling of four angular momentum vectors
Symmetry relations
Reduction to 6j symbols
Special case
Orthogonality relation
3n-j symbols
See also
References
External links

In physics, Wigner's 9-j symbols were introduced by Eugene Paul Wigner in 1937. They are related to recoupling coefficients in quantum mechanics involving four angular momenta

Recoupling of four angular momentum vectors

Coupling of two angular momenta and is the construction of simultaneous eigenfunctions of and , where , as explained in the article on Clebsch–Gordan coefficients.

Coupling of three angular momenta can be done in several ways, as explained in the article on Racah W-coefficients. Using the notation and techniques of that article, total angular momentum states that arise from coupling the angular momentum vectors , , , and may be written as

Alternatively, one may first couple and to and and to , before coupling and to :

Both sets of functions provide a complete, orthonormal basis for the space with dimension spanned by

Hence, the transformation between the two sets is unitary and the matrix elements of the transformation are given by the scalar products of the functions.

As in the case of the Racah W-coefficients the matrix elements are independent of the total angular momentum projection quantum number ():

Symmetry relations

A 9-j symbol is invariant under reflection about either diagonal as well as even permutations of its rows or columns:

An odd permutation of rows or columns yields a phase factor , where

For example:

Reduction to 6j symbols

The 9-j symbols can be calculated as sums over triple-products of 6-j symbols where the summation extends over all {{math|x}} admitted by the triangle conditions in the factors:

.

Special case

When the 9-j symbol is proportional to a 6-j symbol:

Orthogonality relation

The 9-j symbols satisfy this orthogonality relation:

The triangular delta {{math|{j₁ j₂ j₃}}} is equal to 1 when the triad (j₁, j₂, j₃) satisfies the triangle conditions, and zero otherwise.

3n-j symbols

The 6-j symbol is the first representative, {{math|n {{=}} 2}}, of {{math|3n}}-j symbols that are defined as sums of products of {{math|n}} of Wigner's 3-jm coefficients. The sums are over all combinations of {{math|m}} that the {{math|3n}}-j coefficients admit, i.e., which lead to non-vanishing contributions.

If each 3-jm factor is represented by a vertex and each j by an edge, these {{math|3n}}-j symbols can be mapped on certain 3-regular graphs with {{math|3n}} vertices and {{math|2n}} nodes. The 6-j symbol is associated with the K₄ graph on 4 vertices, the 9-j symbol with the utility graph on 6 vertices (K_3,3), and the two distinct (non-isomorphic) 12-j symbols with the Q₃ and Wagner graphs on 8 vertices.

Symmetry relations are generally representative of the automorphism group of these graphs.

References

{{cite book |last1= Biedenharn |first1= L. C. |last2= van Dam |first2= H.

|title= Quantum Theory of Angular Momentum: A collection of Reprints and Original Papers
|year= 1965 |publisher= Academic Press |location= New York |isbn= 0120960567}}

{{cite book |last= Edmonds |first= A. R. |title= Angular Momentum in Quantum Mechanics |year= 1957

|publisher= Princeton University Press |location= Princeton, New Jersey |isbn= 0-691-07912-9}}

{{cite book |last= Condon |first= Edward U. |author2= Shortley, G. H. |title= The Theory of Atomic Spectra |year= 1970

|publisher= Cambridge University Press |location= Cambridge |isbn= 0-521-09209-4 |chapter= Chapter 3}}

{{dlmf|id=34 |title=3j,6j,9j Symbols|first=Leonard C.|last= Maximon}}
{{cite book |last= Messiah |first= Albert |title= Quantum Mechanics (Volume II) |year= 1981 | edition= 12th

|publisher= North Holland Publishing |location= New York |isbn= 0-7204-0045-7}}

{{cite book |last= Brink |first= D. M. |author2= Satchler, G. R. |title= Angular Momentum

|year= 1993 |edition= 3rd |publisher= Clarendon Press |location= Oxford |isbn= 0-19-851759-9 |chapter= Chapter 2 }}

{{cite book |last= Zare |first= Richard N. |title= Angular Momentum |year=1988

|publisher= John Wiley |location= New York |isbn= 0-471-85892-7 |chapter= Chapter 2}}

{{cite book |last= Biedenharn |first= L. C. |author2= Louck, J. D. |title= Angular Momentum in Quantum Physics

|year= 1981 |publisher= Addison-Wesley |location= Reading, Massachusetts |isbn= 0201135078 }}

{{cite book |last= Varshalovich |first= D. A. |author2= Moskalev, A. N.|author3= Khersonskii, V. K.

|title= Quantum Theory of Angular Momentum | year= 1988 |publisher= World Scientific |location= Singapore |isbn= 9971-50-107-4 }}

{{cite journal

|first1=H. A.
|last1=Jahn
|first2=J.
|last2=Hope
|title=Symmetry properties of the Wigner 9j symbol
|journal=Physical Review
|year=1954
|volume=93
|issue=2
|page=318
|doi=10.1103/PhysRev.93.318
|bibcode=1954PhRv...93..318J
}}

External links

{{cite web

|first1=Anthony
|last1=Stone
|url=http://www-stone.ch.cam.ac.uk/wigner.shtml
|title=Wigner coefficient calculator

}} (Gives answer in exact fractions)

{{ cite web

|url=http://plasma-gate.weizmann.ac.il/369j.html
|title=369j-symbol calculator
|author=Plasma Laboratory of Weizmann Institute of Science

}} (Answer as floating point numbers)

{{cite web

|url=http://caagt.ugent.be/yutsis/GYutsisApplet.caagt
|title=GYutsis Applet
|first1=Veerl
|last1=Fack
|first2=Dries
|last2=van Dyck
}}

{{cite web

}} (accurate; C, fortran, python)

{{cite web

|first1=H.T.
|last1=Johansson
|title=(FASTWIGXJ)
|url=http://fy.chalmers.se/subatom/fastwigxj/

}} (fast lookup, accurate; C, fortran)

3 : Rotational symmetry|Representation theory of Lie groups|Quantum mechanics

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