The Wigner-Eckart Theorem



Rotations, spherical tensor operators and the Wigner-Eckart Theorem

From A Web-Based Quantum Mechanics Course with In-Class Tutorials by Marianne Breinig The University of Tennessee, Department of Physics and Astronomy

; this module is:



1. Adding two angular momenta

Vectors in the subspace E(j1,j2) may be expanded in terms of [pic]or [pic]basis vectors.  The basis vectors of one basis can be written as linear combinations of basis vectors of the other basis.

[pic]

[pic].

The expansion coefficients [pic]are called the Clebsch-Gordan coefficients.  Sometimes the Clebsch-Gordan coefficients are written in terms of the Wigner 3-j symbols.  We have

[pic]

Properties of the Wigner 3-j symbols:

[pic]

We can permute the columns of the 3-j symbol.  An even permutation does not alter its value.  An odd permutation multiplies the initial value by [pic].  Moreover,

[pic].

This implies

[pic]

or

[pic]

Adding more than two angular momenta

When adding the angular momenta j1, j2, and j3, many different states of the system may correspond to the same value of j and m.

[pic]

Let [pic]

Then

[pic]

The Racah W coefficients and 6-j symbols transform from one scheme of adding three angular momenta to another.

The 9-j symbols transform from one scheme of adding four angular momenta to another.

Links:

|[p|Racah coefficients, 6-j symbols, 9-j symbols |

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|[p|3-j, 6-j, and 9-j symbol calculators (1) |

|ic| |

|] | |

|[p|3-j, 6-j, and 9-j symbol calculators (2) |

|ic| |

|] | |

[pic]

We will now establish a connection between the Clebsch-Gordan coefficients and the matrix elements of operators that transform like the spherical harmonics under rotation.

The rotation matrices

Angular momentum and rotations are closely linked.  Let us, for a moment, return to the rotation matrices.  An arbitrary rotation can be accomplished in three steps, known as Euler rotations, and it is therefore characterized by three angles, known as Euler angles.  We may proceed as follows.

First rotate the body ccw about the z-axis by an angle α.  Then rotate ccw about the y’-axis by an angle β.  Finally rotate ccw about the z’-axis by an angle γ.

[pic]  [pic]  [pic]

The rotation matrix is R(α,β,γ)=Rz’(γ)Ry’(β)Rz(α).  (Note: In quantum mechanics it is advantageous to perform the second rotation about the y’-axis.  In classical mechanics the second rotation is often performed about the x’-axis.)  Rz’(γ) and Ry’(β) are rotations about body-fixed axis.  We want to express these rotations in terms of rotations about space-fixed axis.  We may write Ry’(β)=Rz(α)Ry(β)Rz-1(α). Similarly, Rz’(γ)=Ry’(β)Rz(γ)Ry’-1(β). We can now rewrite R(α,β,γ) as

R(α,β,γ)=Rz’(γ)Ry’(β)Rz(α). 

=Ry’(β)Rz(γ)Ry’-1(β)Ry’(β)Rz(α)

=Ry’(β)Rz(γ)Rz(α)

=Rz(α)Ry(β)Rz-1(α)Rz(γ)Rz(α)

=Rz(α)Ry(β)Rz(γ).  Rotations about the same axis commute.)

The rotation operator U(R(α,β,γ)) therefore may be written as

U(R(α,β,γ))=U(Rz(α))U(Ry(β))U(Rz(γ)).

It only involves rotations about space-fixed axis.  We already know the rotation operators for rotations about space-fixed axes. (See notes.)

For a spin ½ particle U(R(α,β,γ))=U(Rz(α))U(Ry(β))U(Rz(γ)) becomes

[pic]

[pic]

using

[pic]

[pic].

(See notes.)

The matrix elements of the second rotation are purely real.  This is a result of choosing the y’-axis for our second Euler rotation.

[pic]

For a system with angular momentum J the matrix elements of the rotation operator in the subspace E(j) in the {|j,m>} basis for a rotation R(n,θ) are 

Uj m',m(R)= ................
................

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